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Scale Drawing that shows a real object with accurate sizes reduced or enlarged by a certain amount (called the scale).

The scale is shown as the length in the drawing, then a colon (":"), then the matching length on the real thing.

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The longest side length of a right triangle found directly across from the right angle.

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The side across (opposite) to the reference angle.

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The value multiplied by the pre-image (old image) side lengths to get the image (new image) side lengths.

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How to determine the scale factor of a dilation for enlargement

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A proportion is two fractions that are equal to each other. Similar figures generate multiple proportions as all corresponding side lengths are related by the same scale factor.

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The side length touching the reference angle that is not the hypotenuse.

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Triangles that are related by a dilation. Corresponding side lengths are proportional and corresponding angles are congruent.

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Given two triangles, if we know that two angles are congruent, we know the triangles are similar.

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For right triangles ONLY, this is the ratio of the adjacent side divided by the hypotenuse.

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For right triangles ONLY, this is the ratio of the opposite side divided by the adjacent side.

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For right triangles ONLY, this is the ratio of the opposite side divided by the hypotenuse.

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a^2+b^2=c^2 for any right triangle

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How to copy an angle with compass and straightedge or ruler - Math Open Reference

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Given an angle formed by two lines with a common vertex, this page shows how to construct another angle from it that has the same angle measure using a compass and straightedge or ruler. It works by creating two congruent triangles. A proof is shown below. A Euclidean construction

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How to bisect an angle with compass and straightedge or ruler. To bisect an angle means that we divide the angle into two equal ( congruent) parts without actually measuring the angle. This Euclidean construction works by creating two congruent triangles. See the proof below for more on this.

Printable step-by-step instructions

The above animation is available as a printable step-by-step instruction sheet, which can be used for making handouts or when a computer is not available.

Proof

This construction works by effectively building two congruent triangles. The image below is the final drawing above with the red lines added and points A,B,C labelled.

Try it yourself

Click here for a printable worksheet containing three angle bisection problems. When you get to the page, use the browser print command to print as many as you wish. The printed output is not copyright.

Other constructions pages on this site

Lines

Angles

Triangles

Right triangles

Triangle Centers

Circles, Arcs and Ellipses

Polygons

Non-Euclidean constructions

(C) 2011 Copyright Math Open Reference.
All rights reserved

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How to bisect an angle with compass and straightedge or ruler - Math Open Reference

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How to bisect an angle with compass and straightedge or ruler. To bisect an angle means that we divide the angle into two equal (congruent) parts without actually measuring the angle. This Euclidean construction works by creating two congruent triangles. See the proof below for more on this.

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This construction shows how to draw the perpendicular bisector of a given line segment with compass and straightedge or ruler. This both bisects the segment (divides it into two equal parts, and is perpendicular to it. It finds the midpoint of the given line segment.

Printable step-by-step instructions

The above animation is available as a printable step-by-step instruction sheet, which can be used for making handouts or when a computer is not available.

Proof

This construction works by effectively building congruent triangles that result in right angles being formed at the midpoint of the line segment. The proof is surprisingly long for such a simple construction.

The image below is the final drawing above with the red lines and dots added to some angles.

1

Line segments AP, AQ, PB, QB are all congruent

The four distances were all drawn with the same compass width c.

Next we prove that the top and bottom triangles are isosceles and congruent

Then we prove that the left and right triangles are isosceles and congruent

6

∆APB and ∆AQB are isosceles

Two sides are congruent (length c)

Then we prove that the four small triangles are congruent and finish the proof

12

Each of the four angles at J are 90°. Therefore AB is perpendicular to PQ

They are equal in measure and add to 360°

13

Line segments PJ and QJ are congruent. Therefore AB bisects PQ.

From (8), CPCTC. Corresponding parts of congruent triangles are congruent

Try it yourself

Click here for a printable worksheet containing three bisection problems. When you get to the page, use the browser print command to print as many as you wish. The printed output is not copyright.

Other constructions pages on this site

Lines

Angles

Triangles

Right triangles

Triangle Centers

Circles, Arcs and Ellipses

Polygons

Non-Euclidean constructions

(C) 2011 Copyright Math Open Reference.
All rights reserved

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How to bisect a segment with compass and straightedge or ruler - Math Open Reference

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This construction shows how to draw the perpendicular bisector of a given line segment with compass and straightedge or ruler. This both bisects the segment (divides it into two equal parts), and is perpendicular to it. Finds the midpoint of a line segmrnt. The proof shown below shows that it works by creating 4 congruent triangles.

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LT 4 - Perpendicular line through a point

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LT 4 - Perpendicular line through a point

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This page shows how to construct a perpendicular to a line through an external point, using only a compass and straightedge or ruler. It works by creating a line segment on the given line, then bisecting it. The bisector will be a right angles to the given line. (See proof below).

Printable step-by-step instructions

The above animation is available as a printable step-by-step instruction sheet, which can be used for making handouts or when a computer is not available.

Proof

The image below is the final drawing above with the red lines added.

1

Segment RP is congruent to RQ

They were both drawn with the same compass width

2

Segment SQ is congruent SP

They were both drawn with the same compass width

3

Triangle RQS is congruent to triangle RPS

Three sides congruent (sss), RS is common to both.

4

Angle JRQ is congruent to JRP

CPCTC. Corresponding parts of congruent triangles are congruent.

5

Triangle RJQ is congruent to triangle RJP

Two sides and included angle congruent (SAS), RJ is common to both.

6

Angle RJP and RJQ are congruent

CPCTC. Corresponding parts of congruent triangles are congruent.

7

Angle RJP and RJQ are 90°

They are congruent and supplementary (add to 180°).

Try it yourself

Click here for a printable construction worksheet containing two 'perpendiculars through a point' problems to try. When you get to the page, use the browser print command to print as many as you wish. The printed output is not copyright.

Other constructions pages on this site

Lines

Angles

Triangles

Right triangles

Triangle Centers

Circles, Arcs and Ellipses

Polygons

Non-Euclidean constructions

(C) 2011 Copyright Math Open Reference.
All rights reserved

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How to construct (draw) a perpendicular from a line through an external point - Math Open Reference

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This page shows how to construct a perpendicular to a line through an external point, using only a compass and straightedge or ruler. It works by creating a line segment on the given line, then bisecting it. A Euclidean construction.

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This page shows how to construct a line parallel to a given line that passes through a given point with compass and straightedge or ruler. It is called the 'angle copy method' because it works by using the fact that a transverse line drawn across two parallel lines creates pairs of equal corresponding angles. It uses this in reverse - by creating two equal corresponding angles, it can create the parallel lines.

See also:

Printable step-by-step instructions

The above animation is available as a printable step-by-step instruction sheet, which can be used for making handouts or when a computer is not available.

Proof

This construction works by using the fact that a transverse line drawn across two parallel lines creates pairs of equal corresponding angles. It uses this in reverse - by creating two equal corresponding angles, it can create the parallel lines.

The image below is the final drawing above with the red items added.

1

Line segments AR,BJ are congruent

Both drawn with the same compass width.

2

Line segments RS,JC are congruent

Both drawn with the same compass width.

3

Line segments AS,BC are congruent

Both drawn with the same compass width.

4

Triangles ∆ARS and ∆BJC are congruent

Three sides congruent (sss).

5

Angles ARS, BJC are congruent.

CPCTC. Corresponding parts of congruent triangles are congruent

6

The line AJ is a transversal

It is a straight line drawn with a straightedge and cuts across the lines RS and PQ.

7

Lines RS and PQ are parallel

Angles ARS, BJC are corresponding angles that are equal in measure only if the lines RS and PQ are parallel

Try it yourself

Click here for a printable parallel line construction worksheet containing two problems to try. When you get to the page, use the browser print command to print as many as you wish. The printed output is not copyright.

Other constructions pages on this site

Lines

Angles

Triangles

Right triangles

Triangle Centers

Circles, Arcs and Ellipses

Polygons

Non-Euclidean constructions

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How to construct a parallel through a point with compass and straightedge or ruler - Math Open Reference

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This page shows how to construct a line parallel to a given line that passes through a given point with compass and straightedge or ruler. It is called the 'angle copy method' because it works by using the fact that a transverse line drawn across two parallel lines creates pairs of equal corresponding angles.

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When a figure is inscribed, it is completely inside another figure so that their sides, edges, vertices, or curves touch.

A polygon is inscribed in a circle if it fits inside the circle and every vertex of the polygon is on the circle.

A circle is inscribed in a polygon if it fits inside the polygon and every side of the polygon is tangent to the circle.

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A circle of radius r with center O is the set of all points that are a distance r units from O.

To draw a circle of radius 3 units and center O, use a compass to draw all the points at a distance 3 units from O.

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A line segment is the set of all points on a line between two endpoints.

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Two lines that never intersect are called parallel. Line segments can also be parallel if they extend into parallel lines.

This figure shows two parallel line segments.

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A conjecture is a reasonable guess that someone is trying to either prove or disprove.

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The perpendicular bisector of a segment is a line through the midpoint of the segment that is perpendicular to the segment.

In this diagram, the dashed line is the perpendicular bisector of segment .

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Translate OBJECT by Directed Line Segment from _________ to _____________.

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Rotate OBJECT _______ degrees with center _______.

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Reflect OBJECT over line ___________.

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Angle - Side - Angle Triangle congruence

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Angle - Side - Angle Triangle congruence