Pythagorean Theorem and Its Converse: 8-1 Practice

The Pythagorean Theorem is a mathematical equation that states the sum of the squares of the two shorter sides of a right triangle is equal to the square of the longest side, or hypotenuse. This theorem can be used to easily calculate the length of the hypotenuse or the lengths of the other two sides given a set of measurements. The converse of the Pythagorean Theorem states that if the sum of the squares of two sides of a triangle is equal to the square of the third side, then the triangle is a right triangle. Here we will look at 8-1 practice for the Pythagorean Theorem and its converse.

Using the Pythagorean Theorem

The simplest way to use the Pythagorean Theorem is to calculate the length of the hypotenuse when the lengths of the other two sides are known. For example, if the two sides of a right triangle are 8 cm and 6 cm, then the length of the hypotenuse can be calculated using the Pythagorean Theorem. The equation is a2 + b2 = c2, where a and b are the lengths of the two sides and c is the length of the hypotenuse. In this example, a2 + b2 = 82 + 62 = 64 + 36 = 100. The square root of 100 is 10, so the length of the hypotenuse is 10 cm.

Using the Converse of the Pythagorean Theorem

The converse of the Pythagorean Theorem can be used to determine if a triangle is a right triangle when the lengths of all three sides are known. For example, if the three sides of a triangle are 6 cm, 8 cm, and 10 cm, then the converse of the Pythagorean Theorem can be used to determine if the triangle is a right triangle. The equation is a2 + b2 = c2, where a and b are the lengths of the two shorter sides and c is the length of the longest side. In this example, a2 + b2 = 62 + 82 = 36 + 64 = 100. Since the square of the longest side (102 = 100) is equal to the sum of the squares of the two shorter sides, the triangle is a right triangle.

8-1 Practice

To practice using the Pythagorean Theorem and its converse, try solving the following problems. For each problem, determine if the triangle is a right triangle, and if so, calculate the length of the hypotenuse.

• a=5 cm, b=12 cm, c=13 cm
• a=4 cm, b=3 cm, c=5 cm
• a=3 cm, b=4 cm, c=6 cm

For the first problem, a2 + b2 = 52 + 122 = 25 + 144 = 169. Since 169 is not equal to the square of the longest side (132 = 169), the triangle is not a right triangle. For the second problem, a2 + b2 = 42 + 32 = 16 + 9 = 25. Since 25 is equal to the square of the longest side (52 = 25), the triangle is a right triangle and the length of the hypotenuse is 5 cm. For the third problem, a2 + b2 = 32 + 42 = 9 + 16 = 25. Since 25 is equal to the square of the longest side (62 = 25), the triangle is a right triangle and the length of the hypotenuse is 6 cm.

The Pythagorean Theorem and its converse are two important mathematical equations that can be used to determine the lengths of sides of a right triangle. With 8-1 practice, you can apply these equations to solve problems involving right triangles. With a bit of practice, you will be able to easily calculate the length of the hypotenuse or determine if a triangle is a right triangle.