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# Special Right Triangles

## Key Concepts

• Find hypotenuse length in a 30°-60° -90° triangle

### Special Right Triangles

There are two special right triangles with angles measures as 45°, 45°, 90° degrees and 30°, 60°, 90° degrees.

### 30°-60°-90° Triangle Theorem

In a 30°-60°-90° triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is √3 times as long as the shorter leg.

Hypotenuse = 2 × shorter leg

Longer leg = shorter leg × √3

### Find hypotenuse length in a 30°-60°-90° triangle

Example 1:

Find the height of an equilateral triangle.

Solution:

Draw the equilateral triangle described. Its altitude forms the longer leg of two 30°-60°-90° triangles. The length h of the altitude is approximately the height of the triangle.

longer leg = shorter leg × √3

h = 2 ×√3

h = 2√3

Example 2:

Find the height of an equilateral triangle.

Solution:

Draw the equilateral triangle described. Its altitude forms the longer leg of two 30°-60°-90° triangles.

The length h of the altitude is approximately the height of the triangle.

longer leg = shorter leg ×√3

h = 4×√3

h = 4√3

Example 3:

Find lengths in a 30°-60°-90° triangle. Find the values of x and y. Write your answer in the simplest radical form.

Solution:

STEP 1:

Find the value of x.

longer leg = shorter leg ×√3 (30°-60°-90° Triangle Theorem)

X =√6 ×√3

X =√6 ×√3

X=√18

X=√9 ×2

X= 3√3

STEP 2:

Find the value of y.

hypotenuse =2 × shorter leg               (30°-60°-90° Triangle Theorem)

y =2x √6

y =2√6

Example 4:

A ramp is used to launch a kayak. What is the height of a 10-foot ramp when its angle is 30° as shown?

Solution:

When the body is raised 30°, the height h is the length of the longer leg of a 30°-60°-90° triangle. The length of the hypotenuse is 10 feet.

Hypotenuse = 2 × shorter leg            (30° -60° -90° Triangle Theorem)

10 = 2 × s (Substitute)

5 = s (Divide both sides by 2)

longer leg = shorter leg ×√3

h = 5√3 (Substitute)

h   ≈ 8.5                             (Use a calculator to approximate)

When the angle is 30°, the height of the foot ramp is 8.5 feet.

## Exercise

• Determine the value of the variable.
• Special right triangles: Copy and complete the table.
• A worker is building a ramp of 30° 8 ft to make the transportation of materials easier between an upper and lower platform. The upper platform is 8 feet off the ground, and the angle of elevation is 30°. The ramp length is
• Determine the height of an equilateral triangle.
• Determine the height of an equilateral triangle.

### What have we learned

• Identify special right triangles
• Understand 30° – 60° – 90° triangle thermos
• Understand how to find the height of an equilateral triangle
• Understand how to find lengths in a 30°-60°-90° triangle
• Understand how to find a height

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