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https://www.youtube.com/watch?v=CkbClyVxRIU

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What is a Quadrilateral? – Geometric Shapes – Geometry

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https://byjus.com/maths/quadrilateral/

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A quadrilateral is a closed shape and a type of polygon that has four sides, four vertices and four angles. It is formed by joining four non-collinear points. The sum of interior angles of quadrilaterals is always equal to 360 degrees.

The word quadrilateral is derived from the Latin words ' Quadra' which means four and ' Latus ' means 'sides'. It is not necessary that all the four sides of a quadrilateral are equal in length. Hence, we can have different types of quadrilaterals based on sides and angles. Let us more interesting facts about quadrilaterals in this article.

What is a Quadrilateral?

A quadrilateral is a plane figure that has four sides or edges, and also has four corners or vertices. The angles are present at the four vertices or corners of the quadrilateral. If ABCD is a quadrilateral then angles at the vertices are ∠A, ∠B, ∠C and ∠D. The sides of a quadrilateral are AB, BC, CD and DA.

If we join the opposite vertices of the quadrilateral, we get the diagonals. In the below figure AC and BD are the diagonals of quadrilateral ABCD.

Quadrilaterals will typically be of standard shapes with four sides like rectangle, square, trapezoid, and kite or irregular and uncharacterized as shown below:

Let's have a look at the table below to understand what quadrilateral is.

Types of Quadrilaterals

The types of quadrilaterals are defined based on the measure of the angles and lengths of their sides. As the word 'Quad' means four, all these types of a quadrilateral have four sides, and the sum of angles of these shapes is 360 degrees. The list of types of quadrilaterals are:

Convex, Concave and Intersecting Quadrilaterals

Another way to classify the types of quadrilaterals are:

1. Convex Quadrilaterals: Both the diagonals of a quadrilateral are completely contained within a figure.

2. Concave Quadrilaterals: At least one of the diagonals lies partly or entirely outside of the figure.

3. Intersecting Quadrilaterals: Intersecting quadrilaterals are not simple quadrilaterals in which the pair of non-adjacent sides intersect. These kinds of quadrilaterals are known as self-intersecting or crossed quadrilaterals

Below are the examples of convex, concave and intersecting quadrilaterals.

Properties of Quadrilaterals

Let us understand in a better way with the help of an example:

It has four sides: AB, BC, CD, and DA

It has four vertices: Points A, B, C, and D

It has four angles: ∠ABC, ∠BCD, ∠CDA, and ∠DAB

∠A and ∠B are adjacent angles

∠A and ∠C are the opposite angles

AB and CD are the opposite sides

AB and BC are the adjacent sides

A quadrilateral is a 4-sided plane figure. Below are some important properties of quadrilaterals :

Every quadrilateral has 4 vertices, 4 angles, and 4 sides

The total of its interior angles = 360 degrees

Square Properties

All the sides of the square are of equal measure

The sides are parallel to each other

All the interior angles of a square are at 90 degrees (i.e., right angle)

The diagonals of a square perpendicular bisect each other

To learn more about square, click here.

Rectangle Properties

The opposite sides of a rectangle are of equal length

The opposite sides are parallel to each other

All the interior angles of a rectangle are 90 degrees.

The diagonals of a rectangle bisect each other.

To understand more about rectangle, visit here.

Rhombus Properties

All the four sides of a rhombus are of the same measure

The opposite sides of the rhombus are parallel to each other

The opposite angles are of the same measure

The sum of any two adjacent angles of a rhombus is equal to 180 degrees

The diagonals perpendicularly bisect each other

Learn in detail about rhombus here.

Parallelogram Properties

The opposite side of the parallelogram are of the same length

The opposite sides are parallel to each other

The diagonals of a parallelogram bisect each other

The opposite angles are of equal measure

The sum of two adjacent angles of a parallelogram is equal to 180 degrees

Also, check: Parallelogram

Properties of Trapezium

Only one pair of the opposite side of a trapezium is parallel to each other

The two adjacent sides of a trapezium are supplementary (180 degrees)

The diagonals of a trapezium bisect each other in the same ratio

Click here to learn the definition and formulas related to trapezium.

Properties of Kite

The pair of adjacent sides of a kite are of the same length

The largest diagonal of a kite bisect the smallest diagonal

Only one pair of opposite angles are of the same measure.

A glance at the Quadrilateral Properties

Sides and Angles of Quadrilaterals

Quadrilateral Formulas

There are two basic formulas for quadrilaterals, that are:

Area of Quadrilateral

The area of the quadrilateral is the total space occupied by the figure. The area formula for the different quadrilaterals are given below:

Perimeter of Quadrilateral

Perimeter is the total distance covered by the boundary of a 2d shape. Since we know the quadrilateral has four sides, therefore, the perimeter of any quadrilateral will be equal to the sum of the length of all four sides. If ABCD is a quadrilateral then, the perimeter of ABCD is:

Perimeter = AB + BC + CD + AD

Important Facts of Quadrilateral

A quadrilateral is a trapezoid or a trapezium if 2 of its sides are parallel to each other.

A quadrilateral is a parallelogram if 2 pairs of sides parallel to each other.

Squares and Rectangles are special types of parallelograms. Below are some special properties. - All internal angles are of "right angle" (90 degrees). - Each figure contains 4 right angles. - Sides of a square are of the same length (all sides are congruent) - Opposite sides of a rectangle are the same. - Opposite sides of a rectangle and square are parallel.

A quadrilateral is a rhombus, if
All the sides are of equal length -specified 2 pairs of sides are parallel to each other.

A kite is a special sort of quadrilateral, in which 2 pairs of adjacent sides are equal to each other.

Quadrilaterals Solved Examples

Example 1: What is the base of a rhombus, if its area is 40 square units and the height is 8 units?
Solution:
Given,
Area = 40 square units
Height = 8 units
Area of rhombus = Base × Height
40 = Base × 8
Base = 40/8 = 5 units

Example 2: If 15 metres and 6 metres are diagonal lengths of a kite, then what is its area?

So, the area is simply calculated as, (1/2)(15×6) = 45 m Solution: Given, diagonal 1 = 15 metre and diagonal 2 = 6 metre. ².

Example 3: Find the perimeter of the quadrilateral with sides 5 cm, 7 cm, 9 cm and 11 cm.

Solution: Given, sides of a quadrilateral are 5 cm, 7 cm, 9 cm and 11 cm.

Therefore, perimeter of quadrilateral is:

P = 5 cm + 7 cm + 9 cm + 11 cm = 32 cm

Example 4: The perimeter of the quadrilateral is 50 cm and the lengths of the three sides are 9 cm, 13 cm and 17 cm. Find the missing side of the quadrilateral.

Solution: Let the unknown side of the quadrilateral = x

Given, Perimeter of the quadrilateral = 50 cm

The lengths of other three sides are 9 cm, 13 cm and 17 cm

Perimeter = sum of all the four sides.

50 = 9 cm + 13 cm + 17 cm + x

Therefore, the fourth side of quadrilateral = 11 cm

Practice Questions on Quadrilaterals

1. What is the area of a square with a side length equal to 13 cm?

2. What is the perimeter of rectangle if length = 14cm and breadth = 10 cm?

3. The angles of a quadrilateral are in the ratio 3 : 5 : 9 : 13. Find all the angles of the quadrilateral.

Related Articles

To get more such Math lessons in a simple and easy-to-understand way, register at BYJU'S and score well in your exams.

Frequently Asked Questions

What are Quadrilaterals and their types?

A quadrilateral can be defined as a plane figure having 4 sides. There are mainly 6 types of quadrilaterals which are:

Apart from these 6 types, a quadrilateral can also be classified as:

Concave quadrilaterals

Convex quadrilaterals

Intersecting quadrilaterals

What is a convex and concave quadrilateral?

A convex quadrilateral can be defined as a quadrilateral whose both the diagonals are completely contained within the figure. On the other hand, a concave quadrilateral is a quadrilateral which is having at least one diagonal that lies partly or entirely outside of the figure.

What is the sum of the interior angles of a quadrilateral?

The sum of all the interior angles of a quadrilateral is 360°.

What are the three attributes of a quadrilateral?

The three important attributes of a quadrilateral are:
Four sides
Four Vertices
The sum of the interior angles should be equal to 360 degrees.

How to find the perimeter of a quadrilateral?

The perimeter of a quadrilateral can be determined by adding the side length of all four sides.

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What is Quadrilateral? Properties, Types and Examples of Quadrilaterals

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In geometry, a quadrilateral is a type of polygon that has four sides and four angles. There are different types of quadrilaterals based on their different properties. Learn more about quadrilaterals with examples.

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Vẽ tam giác cân ( hai cạnh bên bằng nhau)

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Vẽ tam giác cân ( hai cạnh bên bằng nhau)

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https://www.youtube.com/watch?v=fd56vYecpqA
Xem video và ghi lại các bước vẽ tam giác cân

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VẼ TAM GIÁC CÂN BẰNG THƯỚC THẲNG VÀ COMPA

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https://im.kendallhunt.com/MS/students/1/1/2/index.html
Game
https://phet.colorado.edu/sims/html/area-builder/latest/area-builder_en.html

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Here are two important principles for finding area:

1. If two figures can be placed one on top of the other so that they match up exactly, then they have the same area.

2. We can decompose a figure (break a figure into pieces) and rearrange the pieces (move the pieces around) to find its area.

Here are illustrations of the two principles.

​

Each square on the left can be decomposed into 2 triangles. These triangles can be rearranged into a large triangle. So the large triangle has the same area as the 2 squares.

Similarly, the large triangle on the right can be decomposed into 4 equal triangles. The triangles can be rearranged to form 2 squares. If each square has an area of 1 square unit, then the area of the large triangle is 2 square units. We also can say that each small triangle has an area of \(\frac12\) square unit.

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Illustrative Mathematics - Students | Kendall Hunt

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Here are two important principles for finding area: If two figures can be placed one on top of the other so that they match up exactly, then they have the same area. We can decompose a figure (break a figure into pieces) and rearrange the pieces (move the pieces around) to find its area.

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https://wheelofnames.com/vi/

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https://youtu.be/hWCtPoOvS0A

Nhiệm vụ 1: Hoạt động ở nhà: Vào link sau thực hiện chơi đủ 6 cấp độ. Chụp hình gửi lại cho tổ trưởng. Tổ trưởng điểm danh.

https://phet.colorado.edu/sims/html/area-builder/latest/area-builder_en.html

Nhiệm vụ 2: Vào link video t rên xem nội dung và ghi lại những nội dung mà em đã học. qua video và ghi lại vào vở chụp nộp tổ trưởng bằng cách trả lời các câu hỏi sau:
1.  Diện tích là gì ( có mấy cách hiểu em ghi hết ra)
2. Đơn vị của diện tích là gì?

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Skip to main content

5.1: Many Dilations of a Triangle (5 minutes)

Warm-up

Students have seen many examples of scale factors that are less than and greater than 1, but the goal of this warm up is to have students focus explicitly on how the size of the scale factor impacts the dilation of a figure. Several dilations of one figure, with the same center but different scale factors, are displayed and students are asked to make observations about the shapes.

If using the digital activity, students will be able to scale a triangle by sliding the scale factor button. Students will informally make conjectures about the relationship between the original and scaled triangles. For example, the triangles are similar, the angles do not change, etc.

Digital

Launch

Show students the image from the task (or demonstrate the applet from the digital activity) and ask "What do you notice? What do you wonder?" Record their responses for all to see. Students may notice and wonder many things. Relevant mathematical things students may notice:

There are triangles that look like dilations of each other.

The center of dilation is point \(P\).

The corresponding sides of the triangles are parallel to each other.

There is a dashed line through each set of corresponding vertices.

The corresponding angles appear to be congruent.

Things students may wonder:

Which was the original triangle, and which are dilations of it?

What scale factors were used?

Give students 2 minutes of quiet work time followed by a whole-class discussion.

Explore the applet and observe the dilation of triangle \(ABC\). The dilation always uses center \(P\), but you can change the scale factor. What connections can you make between the scale factor and the dilated triangle?

Print

Launch

Show students the image from the task (or demonstrate the applet from the digital activity) and ask "What do you notice? What do you wonder?" Record their responses for all to see. Students may notice and wonder many things. Relevant mathematical things students may notice:

There are triangles that look like dilations of each other.

The center of dilation is point \(P\).

The corresponding sides of the triangles are parallel to each other.

There is a dashed line through each set of corresponding vertices.

The corresponding angles appear to be congruent.

Things students may wonder:

Which was the original triangle, and which are dilations of it?

What scale factors were used?

Give students 2 minutes of quiet work time followed by a whole-class discussion.

All of the triangles are dilations of Triangle D. The dilations use the same center \(P\), but different scale factors. What do Triangles A, B, and C have in common? What do Triangles E, F, and G have in common? What does this tell us about the different scale factors used?

Anticipated Misconceptions

If students try to determine the actual scale factors between dilations, let them know that they only need to make an overall observation about the sizes of the dilations compared to Triangle D.

Activity Synthesis

Important observations include

The smaller triangles come from scale factors less than one while the bigger triangles correspond to scale factors larger than one

The original triangle, itself, can be seen as coming from a scale factor of one

All of the triangles have the same angles and orientation in the plane

A few other points worth discussing include

Any of these triangles is a dilation of any other with center \(P\).

The lines from \(P\) contain "corresponding vertices" of the triangles (in the sense that these points map to one another via the dilations taking one triangle to another).

5.2: Info Gap: Dilations (20 minutes)

Activity

This info gap activity gives students an opportunity to determine and request the information needed for a dilation, and to realize that using coordinates greatly simplifies talking about specific points. In order to perform a dilation, students will need to know the center of dilation (which can be communicated using the coordinate grid), the coordinates of the polygon that they are dilating (also communicated using the coordinate grid), and the scale factor. With this information, they can find the dilation as in previous activities.

The info gap structure requires students to make sense of problems by determining what information is necessary, and then to ask for the information they need to solve the problem. This may take several rounds of discussion if their first requests do not yield the information they need (MP1). It also allows them to refine the language they use and ask increasingly more precise questions until they get the information they need (MP6).

Here is the text of one of the cards for reference and planning:

Listen for how students request (and supply) information about the center of dilation and the location of the polygon that is being dilated. The coordinate grid helps name and communicate the location of points, which is essential in this activity. In addition to the location of points, listen for how students use "center of dilation" and "scale factor" in order to communicate this essential information.

Launch

Tell students they will continue to practice describing and drawing dilations using coordinates. Explain the Info Gap and consider demonstrating the protocol if students are unfamiliar with it. Arrange students in groups of 2. Provide access to geometry toolkits. In each group, distribute a problem card to one student and a data card to the other student. After you review their work on the first problem, give them the cards for the second problem and instruct them to switch roles.

Engagement: Develop Effort and Persistence. Display or provide students with a physical copy of the written directions. Check for understanding by inviting students to rephrase directions in their own words. Keep the display of directions visible throughout the activity.
Supports accessibility for: Memory; Organization

Conversing: This activity uses MLR4 Information Gap to give students a purpose for discussing information necessary to dilate a polygon on a coordinate grid. Display questions or question starters for students who need a starting point such as: "Can you tell me . . . (specific piece of information)", and "Why do you need to know . . . (that piece of information)?"
Design Principle(s): Cultivate Conversation

Your teacher will give you either a problem card or a data card. Do not show or read your card to your partner.

If your teacher gives you the problem card:

1. Silently read your card and think about what information you need to be able to answer the question.

2. Ask your partner for the specific information that you need.

3. Explain how you are using the information to solve the problem.

Continue to ask questions until you have enough information to solve the problem.

4. Share the problem card and solve the problem independently.

5. Read the data card and discuss your reasoning.

If your teacher gives you the data card:

1. Silently read your card.

2. Ask your partner "What specific information do you need?" and wait for them to ask for information.

If your partner asks for information that is not on the card, do not do the calculations for them. Tell them you don't have that information.

3. Before sharing the information, ask " Why do you need that information? " Listen to your partner's reasoning and ask clarifying questions.

4. Read the problem card and solve the problem independently.

5. Share the data card and discuss your reasoning.

Pause here so your teacher can review your work. Ask your teacher for a new set of cards and repeat the activity, trading roles with your partner.

Are you ready for more?

Triangle \(EFG\) was created by dilating triangle \(ABC\) using a scale factor of 2 and center \(D\). Triangle \(HIJ\) was created by dilating triangle \(ABC\) using a scale factor of \(\frac12\) and center \(D\).

1. What would the image of triangle \(ABC\) look like under a dilation with scale factor 0?

2. What would the image of the triangle look like under dilation with a scale factor of -1? If possible, draw it and label the vertices \(A'\), \(B'\), and \(C'\). If it's not possible, explain why not.

3. If possible, describe what happens to a shape if it is dilated with a negative scale factor. If dilating with a negative scale factor is not possible, explain why not.

Activity Synthesis

After students have completed their work, share the correct answers and ask students to discuss the process of solving the problems. Some guiding questions:

"Other than the answer, what information would have been nice to have?"

"How did using coordinates help in talking about the problem?"

"If this same problem had a figure on a grid without coordinates, how would you talk about the points?"

"What if there had been no grid at all? Would you still have been able to request or provide the needed information to perform the transformation?"

Highlight that coordinates allow us to unambiguously provide the location of a polygon's vertices. In addition, reinforce the idea that in order to perform a dilation, we need to know the scale factor and the center of dilation. The coordinate grid again provides an efficient means to communicate the center of dilation.

Lesson Synthesis

Ask students to think about the question: "Why does anyone bother putting coordinate axes on a grid? Why are coordinates useful? What are they good for?"

Here are the points to emphasize: Coordinates are an exceptionally powerful tool for communicating the location of points in the plane. There is only one point 3 units to the left of the origin and 2 units up from the origin, the point \((\text-3, 2)\). The location of a polygon is determined by the location and order of its vertices. On a coordinate plane, these can be communicated by giving their coordinates. When we perform a dilation, we also need to know the center of dilation (another point) and the scale factor (a number). On the coordinate plane, all of the information we need to dilate a polygon can be communicated unambiguously with some numbers!

Coordinates in the plane are much like an address in the city: they tell you where to go unambiguously. Because the plane is laid out in a grid, these "addresses" are particularly simple, consisting of two signed numbers. We will use coordinates much more in this unit and beyond, not only to describe individual points but also to describe relationships (like proportional relationships and other new types of relationships).

5.3: Cool-down - Identifying a Dilation (5 minutes)

One important use of coordinates is to communicate geometric information precisely. Let's consider a quadrilateral \(ABCD\) in the coordinate plane. Performing a dilation of \(ABCD\) requires three vital pieces of information:

1. The coordinates of \(A\), \(B\), \(C\), and \(D\)

2. The coordinates of the center of dilation, \(P\)

3. The scale factor of the dilation

With this information, we can dilate the vertices \(A\), \(B\), \(C\), and \(D\) and then draw the corresponding segments to find the dilation of \(ABCD\). Without coordinates, describing the location of the new points would likely require sharing a picture of the polygon and the center of dilation.

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Illustrative Mathematics Grade 8, Unit 2.5 - Teachers | Kendall Hunt

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Skip to main content 5.1: Many Dilations of a Triangle (5 minutes) Warm-up Students have seen many examples of scale factors that are less than and greater than 1, but the goal of this warm up is to have students focus explicitly on how the size of the scale factor impacts the dilation of a figure.

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