## 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 = x^{2} (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.

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