Similar Right Triangles

Key Concepts

• Identify similar triangles

Right angle

the angle bounded by two lines perpendicular to each other: an angle of 90° or ¹/₂ π radians. 

If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other. 

Identify similar triangles 

Example 1: 

Identify the similar triangles in the diagram. 

Solution: 

Sketch the three similar right triangles so that the corresponding angles and sides have the same orientation. 

ΔQRS  ~ ΔPQS ~ Δ PRQ 

Example 2: 

Find the length of the altitude to the hypotenuse. Round decimal answers to the nearest tenth. 

Solution: 

Draw diagram. 

x/23 = 12.8 / 26.6

26.6 (x) = 294.4 

x = 11.1 ft 

Example 3: 

Find the value of y. Write your answer in the simplest radical form. 

Solution: 

Step 1: Draw the three similar triangles 

Step 2: Write a proportion.  

6/x = x/2 (Substitute) 

12 = x² (Cross product property) 

√12 = x (Take the positive square root of each side) 

2√3= x (Simplify) 

Example 4: 

A 30 ft building casts a shadow that is 75 ft long. A nearby tree casts a shadow that is 35 ft long. How tall is the tree?  

Solution: 

𝟑𝟎/𝟕𝟓  =  𝒉/𝟑𝟓          (Corresponding sides of similar figures are proportional) 

75h = 1050   (Find the cross products)  

𝟕𝟓 / 𝒉𝟕𝟓 = 𝟏𝟎𝟓𝟎/𝟕𝟓    (divides both sides by 75) 

 h = 14 

The height of the tree is 14 feet. 

Exercise

• Identify similar triangles. Then find the value of x.

• Charmin is 5.5 feet tall. How far from the wall in the image below would she have to stand in order to measure his height?

• Identifying similar triangles: Identify three similar right triangles in the given diagram.

• Find the length of the altitude to the hypotenuse. Round decimal answers to the nearest tenth.

• Write a similarity statement for the three similar triangles in the diagram. Then complete the proportion.          

• Find the value(s) of the variable(s).

• Using theorems: Tell whether the triangle is a right triangle. If so, find the length of the altitude to the hypotenuse. Round decimal answers to the nearest tenth.

• Describe and correct the error in writing a proportion for the given diagram.

• Finding lengths: Use the Geometric Mean Theorems to find AC and BD.

• Use the diagram. Find FH.

Concept Map

What have we learned

• Identify similar triangles
• Understand how to find the length of the altitude to the hypotenuse
• Understand  geometric mean
• Simplest radical form.
• Understand how to find a height using indirect measurement.