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# 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:

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.

### What have we learned

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

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