Pythagorean Theorem

Key Concepts

• Find the length of a hypotenuse
• Verify right triangles

The Pythagorean Theorem 

In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. 

Find the length of a hypotenuse 

Hypotenuse: A hypotenuse is the longest side of a right-angled triangle, the side opposite the right angle. 

The length of the hypotenuse can be found using the Pythagorean theorem, which states that the square of the length of the hypotenuse equals the sum of the squares of the lengths of the other two sides. 

Example 1: 

Find the length of the hypotenuse of the right triangle. 

Solution: 

(hypotenuse)²   = (leg)² + (leg)²      (Pythagorean Theorem) 

x² = 6² +4² (Substitute) 

x² =36 +16 (Multiply) 

x² = 52 (Add) 

x = (52 )^1/2 (Radical form) 

Example 2: 

Ria ties an apple balloon to a stake in the ground. The rope is 10 feet long. As the wind picks up, Ria observes that the balloon is now 6 feet away from the stake. How far above the ground is the balloon now? 

Solution: 

(Length of rope)²  =  (Distance from stake)²  + (Height of rope)² 

10²  = 6²  + x ² (Substitute)

100 = 36 + x ² (Multiply) 

64 = x ² (Subtract 64 from each side) 

√64  = x (Find positive square root) 

8 = x 

The balloon is 8 ft above the ground. 

Example 3: 

Find the area of the isosceles triangle with side lengths 30 ft., 18 ft., and 18 ft.  

Solution: 

STEP 1: 

Draw a sketch. By definition, the length of an altitude is the height of a triangle. In an isosceles triangle, the altitude to the base is also a perpendicular bisector. So, the altitude divides the triangle into two right triangles with the dimensions shown. 

STEP 2: 

Use the Pythagorean Theorem to find the height of the triangle. 

C² =   a² + b² (Pythagorean Theorem) 

18^{2 =} 15² + h² (Substitute) 

324 = 225 + h² (Multiply)

99 = h² (Subtract 225 from each side)  

9.9=h 

STEP 3: 

Find the area. 

 Area = 1/2 (base) (height) 

            = 1/2 (30) (9.9) 

            = 148.1 ft. ² 

EXAMPLE 3: 

Find the length of the hypotenuse of the right triangle. 

Solution: 

Method 1:  

Use a Pythagorean triple. 

A common Pythagorean triple is 3, 4 , 5.  

Notice that if you multiply the lengths of the legs of the Pythagorean triple by 3, you get the lengths of the legs of this triangle:  

3 × 3 = 9 and 

4 × 3 =12 

So, the length of the hypotenuse is  

5 × 3 = 15 

Method 2: 

Method 2: Use the Pythagorean Theorem. 

 x²   = 9²  +12² 

x²   = 81 + 144 

x²   = 225 

x = 15 

Use the Converse of the Pythagorean Theorem 

The converse of the Pythagorean Theorem is: If the square of the length of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle. 

 Then ΔABC is a right triangle. 

Verify right triangles 

Example 1: 

Write whether the given triangle is a right triangle. 

Solution: 

Let c represent the length of the longest side of the triangle.  

Check to see whether the side lengths satisfy the equation c² = a² + b² ? 

97²   = 65² + 72²? 

9,409 = 4,225 + 5,184 

9,409 = 9,409 

The triangle is a right triangle. 

Exercise

• Find the length of the hypotenuse of the right triangle.

• Find the unknown length of x.

• What is the area of a right triangle with a leg length of 15 feet and a hypotenuse length of 39 feet?

A) 270 ft²  

B) 292.5 ft²

C) 540 ft²   

D) 585 ft²

• Tell whether a triangle with the given side lengths is a right triangle.

3.5, √61

• Tell whether the triangle is a right triangle.

• A student tells you that if you double all the sides of a right triangle, the new triangle is  obtuse. Explain why this statement is incorrect.
• You are making a canvas frame for a painting using stretcher bars. The rectangular shape painting will be 10 inches long and 8 inches wide. With a ruler, how can you be certain that the corners of the frame are 90^°?

• Tell whether the given side lengths of a triangle can represent a right triangle.
• 9, 12, and 15
• 36, 48, and 60
• Decide if the segment lengths form a triangle. Find whether the triangle is acute, right, or obtuse?
• 12, 16, and 20
• 15, 20, and 36
• What type of triangle has side lengths of 4, 7, and 9?

a.  Acute scalene

b.  Right scalene

c.  Obtuse scalene

d.  None of the above

Concept Map

What have we learned

• Understand Pythagorean theorem
• Understand how to find the length of a hypotenuse
• Understand converse of the Pythagorean theorem
• Verify right triangles.