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.

• Determine the value of each variable. Write your answers in the simplest radical form.

• Determine the value of each variable. Calculate your answers in the simplest radical form.

• Detect the values of x and y. Say your answer in the simplest radical form.

• Detect the values of x and y. Tell your answer in the simplest radical form.

• Detect the values of x and y. Say your answer in the simplest radical form.

• 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.

Concept Map

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