 # Similar Triangles and Polygons

A series of free, online High School Geometry Video Lessons and solutions.
Videos, solutions, examples, worksheets, and activities to help Geometry students.

In these lessons, we will learn

• how to solve proportions
• the properties of similar polygons
• how to determine similar triangles
• similar triangles in circles and right triangles (Altitude-on-Hypotenuse Theorems)
• how to use proportions to find an unknown length or distance in similar figures (indirect measurement)

### Solving Proportions

Solving proportions is a crucial skill when studying similar polygons. The ratio of corresponding side lengths between similar polygons are equal and two equivalent ratios are a proportion. For solving proportions problems, we set up the proportions and solve for the missing side length - it will be a variable, or a variable expression

How to solve a simple proportion?
This video demonstrates how to use cross-multiplication to solve simple proportion problems.
Cross Product Property
a/b = c/d
Examples:
a) 5/y = 45/63
b) -1/4 = 3/(2x)
c) (x+3)/12 = 7/2
d) 40/(5x) = (2x)/16
How to solve proportions using cross-multiplication?
Examples:
x/15 = 6/10
3/15 = b/50
20/x = 12/15

### Properties of Similar Polygons

Two polygons are similar if
1. they have the same number of sides.
2. their corresponding angles are congruent.
3. their corresponding sides have a constant ratio (in other words, if they are proportional).

Typically, problems with similar polygons ask for missing sides.
To solve for a missing length, find two corresponding sides whose lengths are known.
After we do this, we set the ratio equal to the ratio of the missing length and its corresponding side and solve for the variable.

How to use the properties of similar polygons to solve for unknown values?
Example:
Given that the two triangles are similar, solve the triangles.
Learn about similar polygons

### Triangle Similarity

There are four triangle congruence shortcuts: SSS, SAS, ASA, and AAS. We have triangle similarity if
(1) two pairs of angles are congruent (AA)
(2) two pairs of sides are proportional and the included angles are congruent (SAS),
or (3) if three pairs of sides are proportional (SSS).
Notice that AAA, AAS, and ASA are not listed -- to include them would be redundant since they all have two congruent angles.
SSS, SAS and AA Triangle Similarity Tutorial

How to determine if two triangles are similar using a shortcut?
Working with similar triangles, determining similar triangles
Examples:
1. The two triangles are similar, Determine the length of the sides DE and EF. Given CB = 15, AC = 9, AB = 10, DF = 6.
2. A tree casts a shadow 45 feet long. At the same time, the shadow cast by a vertical 3 ft stick is 5 ft long. Find the height of the tree.
How to find unknown lengths of sides of similar triangles?
Example:
Find the lengths of the unknown sides given that △MNP ∼ △QSR.
Proportions in Similar Triangles
How to find the missing side of a similar triangle?
How to separate two similar triangles and correctly set up a proportion to solve for one of the missing sides?
In △ABC, D is a point on AB and E is a point on BC such that DE||AC. If DB = 3, AD = 2, and AC = 15, find the length of DE.

### Similar Triangles in Circles and Right Triangles

Two triangles in a circle are similar if two pairs of angles have the same intercepted arc. Sharing an intercepted arc means the inscribed angles are congruent. Since these angles are congruent, the triangles are similar by the AA shortcut. If an altitude is drawn from the right angle in a right triangle, three similar triangles are formed, also because of the AA shortcut.

How to determine if two triangles in a circle are similar and how to prove that three similar triangles exist in a right triangle with an altitude? A lesson on the altitude on hypotenuse theorems
1. The two triangles formed are similar to the right triangle and to each other.

2. The altitude to the hypotenuse is the mean proportional between the segments of the hypotenuse.

3. Either leg of the given right triangle is the mean proportional between the hypotenuse of the given right triangle and the segment of the hypotenuse adjacent to that leg.
A review of problems using the altitude on hypotenuse theorems
Examples:
1. Prove that the product of the measures of the legs of a right triangle is equal to the product of the measures of the hypotenuse and the altitude to the hypotenuse.

2. In the figure, CD is the mean proportional between AD and BD. Find the arithmetic, geometric and harmonic mean between each pair of lengths.

3. Given two positive numbers, a and b, prove that their arithmetic mean, .5(a+b) is always greater than or equal to their positive geometric mean (ab) Altitude-on-Hypotenuse Theorems

### Indirect Measurement

Indirect measurement is a method of using proportions to find an unknown length or distance in similar figures.
Two common ways to achieve indirect measurement involve
(1) using a mirror on the ground and
(2) using shadow lengths and find an object's height.
Method 1 measures the person's height and the distances between the person, mirror, and object.
Method 2 measures shadows and the person's height

How to use similar triangles to measure the height of objects?
Indirect measurement is a brilliant method for measuring distances that would otherwise be difficult or impossible to measure - using the geometrical properties of similar triangles. How to use the properties of similar triangles to determine the height of a tree?

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.  