Pythagorean Theorem

Pythagorean theorem describes the important relationship between the lengths of the sides of a right angled triangle.

Pythagoras theorem states that in a right angled triangle, the square of the length of the hypotenuse c is equal to the sum of the squares of the lengths of the other two legs a and b.

That is, in any right triangle, c²= a²+b².

In any right angled triangle, the side opposite the right angle is called the hypotenuse and this is always the longest side.

Example:

1. Find the length of the hypotenuse in the given triangle.

c²=6²+10² (Pythagorean Theorem)

   = 36 + 100 (simplify)

  =136 (add)

c= √136 (take the square root)

 c =11.7cm

Converse

If a² + b² = c², then the triangle is right-angled triangle.
In other words, if a² + b² = c², a triangle exists with sides a, b and c such that a right angle lies between the sides of length a and b

eg.2. The lengths of the three sides of a triangle are 5, 7, and 9 inches. Determine whether this triangle is a right triangle.

Since the longest side is 9 inches, use 9 as c, the measure of the hypotenuse.

c²=a²+b² (Pythagoras theorem)

Check whether 9²=5²+7²

9²=81

5²+7²=25+49=74

81is not equal to 74.

Since c² ≠ a²+b², the triangle is not a right triangle.

Pythagorean Triples

There are three common relations between the sides of a right triangle. Recognizing these common side lengths can save considerable calculation time. Rather than using the Pythagorean Theorem to calculate the missing side length, the length of the side can be determined by noticing the pattern.

3-4-5 triangles: 3² + 4² = 5²
6-8-10, 9-12-15, and 12-16-20 triangles are simply multiples of the 3-4-5 rule.

5-12-13 triangles: 5 ² + 12 ² = 13 ²
10-24-26 is another common way for this ratio to appear.

8-15-17 triangles: 8² + 15 ² = 17²

Example 3:

If a right triangle has legs 10 and 24, what is the length of the third side?

Instead of performing the calculation to find the third side, recognize that this triangle is a multiple of 5-12-13, where each side of the triangle is double the 5-12-13 pattern. Consequently, the third side is 2(13) = 26.

Choose the correct answer:

1. Use the figure below to answer the question.

The picture shows the dimensions of the roof of a house. What is the height, h to the nearest tenth of a foot?

(a) 3.3		(b) 5.0			(c) 11.0		(d)17.7
 

2. Look at the cone shown below.

Which of the following cones is similar to the one above?