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How do transformations move a figure on the coordinate plane?

How do we use the definition of rigid motion to prove triangle congruency?

How can I make formal constructions using a variety of tools?

How will transformations be used in the coordinate plane?

How can I define congruence in rigid motion?

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Students will perform basic constructions using a compass and straightedge.

Students will inscribe an equilateral triangle, square and regular hexagon in a circle.

Students will classify triangles on the coordinate plane using the Pythagorean Theorem and Distance Formula.

Students will be able to use transformations in the coordinate plane.

Students will be able to perform double transformations in the coordinate plane.

Students will specify a sequence of transformations that will carry a given figure onto another.

Students will understand corresponding parts of congruent triangles are congruent.

Students will prove triangles are congruent using the distance formula, Pythagorean
Theorem, and protractors, if necessary.

Students will prove triangle congruency by SSS, SAS, and ASA using the definition of rigid motion, applying the distance formula and using protractors.

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PRIORITY:
G.CO.5​ Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

G.CO.8​ Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of
congruence in terms of rigid motions.

SUPPORTING:
G.CO.1​ Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment,
based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
G.CO.2​ Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).

G.CO.3​ Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and
reflections that carry it onto itself.

G.CO.4​ Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

G.CO.6​ Use geometric descriptions of rigid motions to transform figures and to predict the effect of a
given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid
motions to decide if they are congruent.

G.CO.7​ Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

G.CO.12​ Make formal geometric constructions with a variety of tools and methods (compass and
straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

G.CO.13​ Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

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Point
Line
Segment
Compass
Straightedge
Angle
Center
Triangle
Square
Hexagon
Circle
Regular
Endpoint
Midpoint
Congruent
Flip
Mirror
Shift
Turn
Move
Trace
Tracing paper
Patty paper
Counter-clockwise
Clockwise
Vertices
Pre-image
Image
Triangle
Same
Equal
Corresponding parts

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Center
Vertex
Bisector
Polygon
Quadrilateral
Isosceles
Scalene
Equilateral
Inscribe
Radius
Diameter
Perpendicular
Parallel
Distance formula
Pythagorean theorem
Triangle inequality
Rotation
Reflection
Translation
Transformation
Prime
Double-prime
Double transformation
Polygons
Origin
Side-Side-Side
Side-Angle-Side
Angle-Side-Angle
Congruent sides
Congruent angles
Side length
Angle measure
Protractor
Geometric symbols
Congruence symbols

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- [Voiceover] What I hope to introduce you to in this video is the notion of a transformation in mathematics, and you're probably used to the word in everyday language. Transformation means something is changing, it's transforming from one thing to another. What would transformation mean in a mathematical context? Well, it could mean that you're taking something mathematical and you're changing it into something else mathematical, that's exactly what it is. It's talking about taking a set of coordinates or a set of points, and then changing them into a different set of coordinates or a different set of points. For example, this right over here, this is a quadrilateral we've plotted it on the coordinate plane. This is a set of points, not just the four points that represent the vertices of the quadrilateral, but all the points along the sides too. There's a bunch of points along this. You could argue there's an infinite, or there are an infinite number of points along this quadrilateral. This right over here, the point X equals 0I equals negative four, this is a point on the quadrilateral. Now, we can apply a transformation to this, and the first one I'm going to show you is a translation, which just means moving all the points in the same direction, and the same amount in that same direction, and I'm using the Khan Academy translation widget to do it. Let's translate, let's translate this, and I can do it by grabbing onto one of the vertices, and notice I've now shifted it to the right by two. Every point here, not just the orange points has shifted to the right by two. This one has shifted to the right by two, this point right over here has shifted to the right by two, every point has shifted in the same direction by the same amount, that's what a translation is. Now, I've shifted, let's see if I put it here every point has shifted to the right one and up one, they've all shifted by the same amount in the same directions. That is a translation, but you could imagine a translation is not the only kind of transformation. In fact, there is an unlimited variation, there's an unlimited number different transformations. So, for example, I could do a rotation. I have another set of points here that's represented by quadrilateral, I guess we could call it CD or BCDE, and I could rotate it, and I rotate it I would rotate it around the point. So for example, I could rotate it around the point D, so this is what I started with, if I, let me see if I can do this, I could rotate it like, actually let me see. So if I start like this I could rotate it 90 degrees, I could rotate 90 degrees, so I could rotate it, I could rotate it like, that looks pretty close to a 90-degree rotation. So, every point that was on the original or in the original set of points I've now shifted it relative to that point that I'm rotating around. I've now rotated it 90 degrees, so this point has now mapped to this point over here. This point has now mapped to this point over here, and I'm just picking the vertices because those are a little bit easier to think about. This point has mapped to this point. The point of rotation, actually, since D is actually the point of rotation that one actually has not shifted, and just 'til you get some terminology, the set of points after you apply the transformation this is called the image of the transformation. So, I had quadrilateral BCDE, I applied a 90-degree counterclockwise rotation around the point D, and so this new set of points this is the image of our original quadrilateral after the transformation. I don't have to just, let me undo this, I don't have to rotate around just one of the points that are on the original set that are on our quadrilateral, I could rotate around, I could rotate around the origin. I could do something like that. Notice it's a different rotation now. It's a different rotation. I could rotate around any point. Now let's look at another transformation, and that would be the notion of a reflection, and you know what reflection means in everyday life. You imagine the reflection of an image in a mirror or on the water, and that's exactly what we're going to do over here. If we reflect, we reflect across a line, so let me do that. This, what is this one, two, three, four, five, this not-irregular pentagon, let's reflect it. To reflect it, let me actually, let me actually make a line like this. I could reflect it across a whole series of lines. Woops, let me see if I can, so let's reflect it across this. Now, what does it mean to reflect across something? One way I imagine is if this was, we're going to get its mirror image, and you imagine this as the line of symmetry that the image and the original shape they should be mirror images across this line we could see that. Let's do the reflection. There you go, and you see we have a mirror image. This is this far away from the line. This, its corresponding point in the image is on the other side of the line but the same distance. This point over here is this distance from the line, and this point over here is the same distance but on the other side. Now, all of the transformations that I've just showed you, the translation, the reflection, the rotation, these are called rigid transformations. Once again you could just think about what does rigid mean in everyday life? It means something that's not flexible. It means something that you can't stretch or scale up or scale down it kind of maintains its shape, and that's what rigid transformations are fundamentally about. If you want to think a little bit more mathematically, a rigid transformation is one in which lengths and angles are preserved. You can see in this transformation right over here the distance between this point and this point, between points T and R, and the difference between their corresponding image points, that distance is the same. The angle here, angle R, T, Y, the measure of this angle over here, if you look at the corresponding angle in the image it's going to be the same angle. The same thing is true if you're doing a translation. You could imagine these are acting like rigid objects. You can't stretch them, they're not flexible they're maintaining their shape. Now what would be examples of transformations that are not rigid transformations? Well you could imagine scaling things up and down. If I were to scale this out where it has maybe the angles are preserved, but the lengths aren't preserved that would not be a rigid transformation. If I were to just stretch one side of it, or if I were to just pull this point while the other points stayed where they are I'd be distorting it or stretching it that would not be a rigid transformation. So hopefully this gets you, it's actually very, very interesting. When you use an art program, or actually you use a lot of computer graphics, or you play a video game, most of what the video game is doing is actually doing transformations. Sometimes in two dimensions, sometimes in three dimensions, and once you get into more advanced math, especially things like linear algebra, there's a whole field that's really focused around transformations. In fact, some of the computers with really good graphics processors, a graphics processor is just a piece of hardware that is really good at performing mathematical transformations, so that you can immerse yourself in a 3D reality or whatever else. This is really really interesting stuff.

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Intro to geometric transformations (video) | Khan Academy

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Voiceover] What I hope to introduce you to in this video is the notion of a transformation in mathematics, and you're probably used to the word in everyday language. Transformation means something is changing, it's transforming from one thing to another. What would transformation mean in a mathematical context?

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Congruence and Transformations

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Transformations & congruence

Transformations & congruence

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Congruence & transformations (practice) | Khan Academy

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Identify Transformations

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Mapping Shapes

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Mapping shapes (practice) | Khan Academy

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Let's say we have triangle ABC. It looks something like this. I want to think about the minimum amount of information. I want to come up with a couple of postulates that we can use to determine whether another triangle is similar to triangle ABC. So we already know that if all three of the corresponding angles are congruent to the corresponding angles on ABC, then we know that we're dealing with congruent triangles. So for example, if this is 30 degrees, this angle is 90 degrees, and this angle right over here is 60 degrees. And we have another triangle that looks like this, it's clearly a smaller triangle, but it's corresponding angles. So this is 30 degrees. This is 90 degrees, and this is 60 degrees, we know that XYZ in this case, is going to be similar to ABC. So we would know from this because corresponding angles are congruent, we would know that triangle ABC is similar to triangle XYZ. And you've got to get the order right to make sure that you have the right corresponding angles. Y corresponds to the 90-degree angle. X corresponds to the 30-degree angle. A corresponds to the 30-degree angle. So A and X are the first two things. B and Y, which are the 90 degrees, are the second two, and then Z is the last one. So that's what we know already, if you have three angles. But do you need three angles? If we only knew two of the angles, would that be enough? Well, sure because if you know two angles for a triangle, you know the third. So for example, if I have another triangle that looks like this-- let me draw it like this-- and if I told you that only two of the corresponding angles are congruent. So maybe this angle right here is congruent to this angle, and that angle right there is congruent to that angle. Is that enough to say that these two triangles are similar? Well, sure. Because in a triangle, if you know two of the angles, then you know what the last angle has to be. If you know that this is 30 and you know that that is 90, then you know that this angle has to be 60 degrees. Whatever these two angles are, subtract them from 180, and that's going to be this angle. So in general, in order to show similarity, you don't have to show three corresponding angles are congruent, you really just have to show two. So this will be the first of our similarity postulates. We call it angle-angle. If you could show that two corresponding angles are congruent, then we're dealing with similar triangles. So for example, just to put some numbers here, if this was 30 degrees, and we know that on this triangle, this is 90 degrees right over here, we know that this triangle right over here is similar to that one there. And you can really just go to the third angle in this pretty straightforward way. You say this third angle is 60 degrees, so all three angles are the same. That's one of our constraints for similarity. Now, the other thing we know about similarity is that the ratio between all of the sides are going to be the same. So for example, if we have another triangle right over here-- let me draw another triangle-- I'll call this triangle X, Y, and Z. And let's say that we know that the ratio between AB and XY, we know that AB over XY-- so the ratio between this side and this side-- notice we're not saying that they're congruent. We're looking at their ratio now. We're saying AB over XY, let's say that that is equal to BC over YZ. That is equal to BC over YZ. And that is equal to AC over XZ. So once again, this is one of the ways that we say, hey, this means similarity. So if you have all three corresponding sides, the ratio between all three corresponding sides are the same, then we know we are dealing with similar triangles. So this is what we call side-side-side similarity. And you don't want to get these confused with side-side-side congruence. So these are all of our similarity postulates or axioms or things that we're going to assume and then we're going to build off of them to solve problems and prove other things. Side-side-side, when we're talking about congruence, means that the corresponding sides are congruent. Side-side-side for similarity, we're saying that the ratio between corresponding sides are going to be the same. So for example, let's say this right over here is 10. No. Let me think of a bigger number. Let's say this is 60, this right over here is 30, and this right over here is 30 square roots of 3, and I just made those numbers because we will soon learn what typical ratios are of the sides of 30-60-90 triangles. And let's say this one over here is 6, 3, and 3 square roots of 3. Notice AB over XY 30 square roots of 3 over 3 square roots of 3, this will be 10. What is BC over XY? 30 divided by 3 is 10. And what is 60 divided by 6 or AC over XZ? Well, that's going to be 10. So in general, to go from the corresponding side here to the corresponding side there, we always multiply by 10 on every side. So we're not saying they're congruent or we're not saying the sides are the same for this side-side-side for similarity. We're saying that we're really just scaling them up by the same amount, or another way to think about it, the ratio between corresponding sides are the same. Now, what about if we had-- let's start another triangle right over here. Let me draw it like this. Actually, I want to leave this here so we can have our list. So let's draw another triangle ABC. So this is A, B, and C. And let's say that we know that this side, when we go to another triangle, we know that XY is AB multiplied by some constant. So I can write it over here. XY is equal to some constant times AB. Actually, let me make XY bigger, so actually, it doesn't have to be. That constant could be less than 1 in which case it would be a smaller value. But let me just do it that way. So let me just make XY look a little bit bigger. So let's say that this is X and that is Y. So let's say that we know that XY over AB is equal to some constant. Or if you multiply both sides by AB, you would get XY is some scaled up version of AB. So maybe AB is 5, XY is 10, then our constant would be 2. We scaled it up by a factor of 2. And let's say we also know that angle ABC is congruent to angle XYZ. I'll add another point over here. So let me draw another side right over here. So this is Z. So let's say we also know that angle ABC is congruent to XYZ, and let's say we know that the ratio between BC and YZ is also this constant. The ratio between BC and YZ is also equal to the same constant. So an example where this 5 and 10, maybe this is 3 and 6. The constant we're kind of doubling the length of the side. So is this triangle XYZ going to be similar? Well, if you think about it, if XY is the same multiple of AB as YZ is a multiple of BC, and the angle in between is congruent, there's only one triangle we can set up over here. We're only constrained to one triangle right over here, and so we're completely constraining the length of this side, and the length of this side is going to have to be that same scale as that over there. And so we call that side-angle-side similarity. So once again, we saw SSS and SAS in our congruence postulates, but we're saying something very different here. We're saying that in SAS, if the ratio between corresponding sides of the true triangle are the same, so AB and XY of one corresponding side and then another corresponding side, so that's that second side, so that's between BC and YZ, and the angle between them are congruent, then we're saying it's similar. For SAS for congruency, we said that the sides actually had to be congruent. Here we're saying that the ratio between the corresponding sides just has to be the same. So for example SAS, just to apply it, if I have-- let me just show some examples here. So let's say I have a triangle here that is 3, 2, 4, and let's say we have another triangle here that has length 9, 6, and we also know that the angle in between are congruent so that that angle is equal to that angle. What SAS in the similarity world tells you is that these triangles are definitely going to be similar triangles, that we're actually constraining because there's actually only one triangle we can draw a right over here. It's the triangle where all the sides are going to have to be scaled up by the same amount. So there's only one long side right here that we could actually draw, and that's going to have to be scaled up by 3 as well. This is the only possible triangle. If you constrain this side you're saying, look, this is 3 times that side, this is 3 three times that side, and the angle between them is congruent, there's only one triangle we could make. And we know there is a similar triangle there where everything is scaled up by a factor of 3, so that one triangle we could draw has to be that one similar triangle. So this is what we're talking about SAS. We're not saying that this side is congruent to that side or that side is congruent to that side, we're saying that they're scaled up by the same factor. If we had another triangle that looked like this, so maybe this is 9, this is 4, and the angle between them were congruent, you couldn't say that they're similar because this side is scaled up by a factor of 3. This side is only scaled up by a factor of 2. So this one right over there you could not say that it is necessarily similar. And likewise if you had a triangle that had length 9 here and length 6 there, but you did not know that these two angles are the same, once again, you're not constraining this enough, and you would not know that those two triangles are necessarily similar because you don't know that middle angle is the same. Now, you might be saying, well there was a few other postulates that we had. We had AAS when we dealt with congruency, but if you think about it, we've already shown that two angles by themselves are enough to show similarity. So why worry about an angle, an angle, and a side or the ratio between a side? So why even worry about that? And we also had angle-side-angle in congruence, but once again, we already know the two angles are enough, so we don't need to throw in this extra side, so we don't even need this right over here. So these are going to be our similarity postulates, and I want to remind you, side-side-side, this is different than the side-side-side for congruence. We're talking about the ratio between corresponding sides. We're not saying that they're actually congruent. And here, side-angle-side, it's different than the side-angle-side for congruence. It's this kind of related, but here we're talking about the ratio between the sides, not the actual measures.

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Triangle similarity postulates/criteria (video) | Khan Academy

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Let's say we have triangle ABC. It looks something like this. I want to think about the minimum amount of information. I want to come up with a couple of postulates that we can use to determine whether another triangle is similar to triangle ABC.

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Determine similar triangles: AA

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Determine similar triangles: Angles (practice) | Khan Academy

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Given two triangles with some of their angle measures, determine whether the triangles are similar or not.

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Determine similar triangles: SSS

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Determine similar triangles: SSS (practice) | Khan Academy

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Given two triangles with their side measures, determine whether the triangles are similar or not.

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After any of those transformations (turn, flip or slide), the shape still has the same size, area, angles and line lengths.

When one shape can become another using only Turns, Flips and/or Slides, then the two shapes are Congruent

When a Resize is neeeded for one shape to become another (we may also Turn, Flip and/or Slide) the two shapes are

In other words:

When one shape becomes another ...

The shapes are ...

... using only Rotate, Reflect and/or Translate

... using a Resize (may also Rotate, Reflect
and/or Translate)

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Learn about the Four Transformations: Rotation, Reflection, Translation and Resizing

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Transformations and Symmetry

The word "transformation" refers to a movement or change in a geometric shape. If we slide a geometric shape to the right or left, this is called a "translation". In the example below, the original triangle is triangle ABC, and the second triangle A'B'C' is a translation of the original triangle. In moving the first triangle two inches to the right, we say that we translated triangle ABC two inches horizontally to create triangle A'B'C'.

In the second example of translation below, triangle DEF has been translated one inch vertically to create triangle D'E'F':

The second kind of transformation is called a "reflection". When you look in a mirror, you see a reflection of yourself, but left and right are reversed. As you see in the diagram below, when we reflect triangle ABC, the reflected triangle A'B'C' is "flipped" or reversed. Line "m" is the mirror.

A third kind of transformation is called a "rotation". If you want to rotate an object, you must first choose the Center of Rotation. This point is the point that you will rotate the object around, just as the center of a wheel is the center about which the wheel rotates. In the diagram below, triangle DEF has been rotated 90 degrees counterclockwise to its new position, triangle D'E'F'. P is the center of rotation.

In the diagram below, a triangle has been rotated 6 times about point P, at 60 degrees each time. Since 6 times 60 is 360 degrees, we see that this completes a full circle:

The fourth transformation is called "dilation". When the eye doctor "dilates your eyes" during an eye examination, the pupils of your eyes become enlarged so that the doctor can see the veins of your eyes more clearly. Dilation, mathematically, means to take a shape and make it larger or smaller but keep it the same proportion. In the geometric example below, the triangle on the left is "dilated" by a factor of 2 (it is "doubled") to create the dilated triangle on the right. Each side of the new triangle is twice as long as the corresponding side of the original triangle. Point "P" is the center of dilation. We would therefore say that the "scale factor is 1 to 2" comparing the original to the new figure. Another way to compare the sizes of the figures is to say that the smaller triangle is "one-half the size" of the larger figure".

What do you think is true of the angles of the new triangle compared to the angles of the original triangle? If you answered "they are the same", or "the new angles are the same measure, or number of degrees, as the original triangle" then you are correct! Since the angles are the same, the two triangles are the same shape.

You will learn more about transformations and symmetry at the following web page:

http://mathforum.com/sum95/suzanne/symsusan.html

Test Question #1: What is special about the word ATOYOTA, as explained on the web site linked above?

To create an interesting geometric, you can combine the two or three of the transformations (translation, reflection, rotation) and use a variety of shapes and colors as in the example below. Which of the transformations were used?

If you answered "reflection and rotation", then you are correct!

Translation and dilation were used to construct the "frieze pattern" below. A "frieze pattern" is a design made from horizontal or vertical repetitions of the same shape.

Frieze patterns are often used in fabric designs for clothing, draperies, and wallpaper, and even in architecture, such as the example below, a Tile Frieze from the Palacio de Velazquez in Madrid, Spain

(photo from http://www.fandm.edu/Departments/Mathematics/a.crannell/hm/Math/seven.html)

You will find a mathematical discussion about these and other transformations at the following web page, on the Math Forum website. The Math Forum is a fascinating web site, with thousands of pages about math.

Test Question #2: What type of transformation is discussed on the Math Forum web page that was NOT discussed on this MathArt page, what is the definition of that type of transformation, and how does it differ from the 3 types of transformation discussed in MathArt?

We say that an object has symmetry, or is symmetrical, if the two sides are "of similar measure". Many animals and insects are symmetrical; a beautiful example of this is the Butterfly below.

We say that a butterfly, and the human body, are symmetrical because the left and right sides appear to be very much alike. This type of symmetry is called "reflection symmetry" because the left side, if it were reflected across the middle line, would match the right side. You can see this general symmetry in the following famous drawing by Leonardo Da Vinci. In this drawing, Da Vinci illustrates the form with two positions of both the arms and legs, depicting the symmetry and proportion of the human figure.

Leonardo Da Vinci is often called the "Renaissance Man", because he combined imagination, curiosity and creativity. He was a scientist, an inventor, and an artist. His most famous painting is The Mona Lisa. The web page below will give you an introduction to the artistic side of Leonardo:

I would like you to read the following web page from the Leonardo web site, but you do not have to do any of the activities (although you will probably find them intriguing):

You will find some interesting information about symmetry at the following web page, on the Da Vinci web site:

http://www.fi.edu/qa97/me3/

Test Question #3: using the method explained in the web page above, "de-code" the following question and answer:

(photo from http://members.aol.com/alohaquilt/index.html)

Many pieces of art work are symmetrical, and so are many graphic designs. Some of the most beautiful examples of symmetry can be found in Hawaiian quilts, such as the one below. You will find out how to create your own geometric quilt pattern at the web page below:

http://www.k12.hi.us/~csanders/Quilt.html

You will find out more information about symmetry by reading the one page linked below, from "The Geometry Pages". You do not have to go to any other pages besides this one page; please come back to the MathArt pages when you have carefully read the page below. Click on this link now:

Test Question #4: Name the two types of symmetry, and give an example (from the above web page) for each type.

From earliest recorded history, we find examples of artists and architects designing their paintings and buildings with a strong emphasis on symmetry. For example, most buildings built before the 20th century had very strong symmetry: the left side of the structure is exactly the same as the right side, as you can see in the images below:

(Photos from http://www.GreatBuildings.com/)

From left to right:

Torii of Itsukushima on Miyajima Island in Japan 10th century AD, Norwegian storehouse 13th century AD, Piazza of St. Peter's in Rome 17th century, S. Maria Novella Church in Florence Italy 15th century AD, The Taj Mahal in Agra India 17th century, United States Capitol in Washington D.C. 19th century.

The following two images are from http://homer.reed.edu/Parthenon.html and the text is from http://www.johnco.cc.ks.us/~jjackson/part.html

"The supreme achievement of the Greeks was their claim that everything--from the human body to the entire cosmos--was governed by an order accessible to human reason. That claim has been the basis for western civilization ever since. The Temple of Athena Parthenos, the Parthenon, now a ruin lying atop the Acropolis of Athens, is the testament in stone to that claim."

And this " order accessible to human reason" is mathematical symmetry! The Parthenon is one of the classical examples of ancient Greek civilization, built in 447 BC. The structure, with beautiful proportions and perfect symmetry, is also an example of the importance of symmetry and proportion, and therefore mathematics, in architecture.

The floor plan of the Parthenon below shows the very strong order and symmetry, and emphasis on geometric shapes that is found in most architecture:

You will find further information at the following web page, called "The Golden Section in Architecture".

This page is rich with information on both mathematics and architecture and you should spend quite a lot of time reading and looking through it. As you scroll down you will also find "The Golden Section in Art", which you should also read. I would like you to go to some of the links from this page, and read what you find there. You will find many good ideas for your project! So what is the "Golden Section"? You will find the answer to this question, and some good project ideas, at the following web page:

As you have seen, many architects design their buildings so that they are symmetrical, and this has been true throughout history. You will also find many examples of artistic use of symmetry in much of the ornamental designs on the walls of many buildings. One very good example is the Alhambra Palace, in Spain. You will find some interesting information about this beautiful building, and examples of the symmetry in the ornamental tiles of the Alhambra by clicking on the link below:

The transformation called "dilation" produces figures that are the same shape, but not necessarily the same size, as explained earlier on this page. In the example above, the original figure was one-half the size of the new figure. This meant that each side of the original triangle was one-half as long as the corresponding ("matching") side of the "dilated" triangle.

This mathematical concept is used extensively in building model airplanes, dollhouses, and in architectural design as well. Certainly an architect cannot possible draw a building at its full size - it would take an enormous sheet of paper! So an architect (or an engineer, or an airplane designer, etc.) draws the building "to scale". The phrase "scale drawing" refers to a drawing that is smaller (or occasionally larger) than the real thing. The floor plan of the house shown below has been drawn at to scale.

The original scale at which this floor plan was created was 1/4 inch represents 1 foot, which means that every 1/4" inch on the drawing represents 1 foot in the real house. (Note: to fit the drawing onto a web page, it has been reproduced, so is not measureable as you see it now.)

Now, some mathematics: if 1/4 inch represents 1 foot, then it would also be true that 1/4 inch represents 12 inches. We can simplify this equation by multiplying both sides by 4 as shown below:

Therefore, the ratio is 1 to 48, and the actual drawing was one forty-eighth the size of the real house.

Architects and engineers write it this way:

To learn more about the mathematics of scale drawing, visit the following page:

Test Question #5: The length of a stadium is 200 yards and its width is 120 yards. If 1 inch represents 10 yards, what would be the dimensions of the stadium drawn on a sheet of paper?

Answer: 20 inches by 12 inches.

In searching on in the internet for information about architects and architecture, I found the following website. It is fascinating! If you are interested in possibly becoming an architect, please visit this website. (It is not a required part of this class, though)
As you have seen, architects, artists, and designers use the mathematical concepts of transformations - translation, reflection, rotation and dilation - in creating their work. You, too, can create beautiful designs using the mathematical concepts of transformations and symmetry. In your project for this chapter, you will design a beautiful geometric graphic. You will find out more about how to create these geometric graphics at the following web page:
http://www.punahou.edu/acad/sanders/geometrypages/GP05Constructions.html
Chapter 2 Project

Your assignment for this project is to create an artistic tessellation, using geometric shapes. First do some research on the Internet, and find some tessellations. Read some information about the tessellations, and explore any aspects that interest you. Then design and create a tessellation.

You must use only geometric shapes in your design (triangles, pentagons, hexagons, circles, half-circles etc.)

Draw the graphic using compass and ruler, in pen on unlined white paper then color it using colored pens or pencils OR construct the graphic using The Geometer's Sketchpad, hide all the points, print it out and then color it with colored pens or pencils. * *

If you use pencil, paper and compass, lay out your design with light pencil lines, then complete the graphic using compass and ruler, in pen, then color it using colored pens

If you use Sketchpad, hide all the points before you print it. (Click on one point. Then use the Edit menu to Select All Points. Then use the Display menu to Hide the points.)

For both of these options, complete your geometric graphic using a black pen for the outlines, and colored pens or colored pencils for shading.

Write at least 30 words about the transformations that you used, the kind of symmetry that your design has, and the names of the geometric figures in your design (for example: "I drew three connected lines on the left side of the rectangle and translated the lines to the right. Then I drew a half-circle and a triangle on the top of the rectangle ..." etc.)

*Please remember that one of your two projects this semester MUST be with paper, pencil, compass and ruler and the other MUST be using GSP. Both of your projects this semester MUST be in at least three colors. If you decide to do this project using The Geometer's Sketchpad computer software, you might need to review the information on the web page called The Geometer's Sketchpad at The Geometer's Sketchpad

Please do not fold the project - you will have spent a lot of time and care creating your design, and you want it to look nice, so please use a large envelope and a piece of cardboard along with the drawing so that it is not bent in the mail.

Mail your project to me:
Mrs. Sanders

1953 Paula Drive

Honolulu, Hawaii 96816
Go To Chapter 3: Tessellations

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MathArt Ch. 2 Transformations and Symmetry

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Transformations and Symmetry The word "transformation" refers to a movement or change in a geometric shape. If we slide a geometric shape to the right or left, this is called a "translation". In the example below, the original triangle is triangle ABC, and the second triangle A'B'C' is a translation of the original triangle.

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Proving triangles congruent

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Unit 1: Congruence, proof, and construction

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Any point on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoint Alternate interior and corresponding angles Congruence, proof, and construction You have seen that two triangles are congruent, if all six pairs of corresponding