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Chapter 9—Basic Geometry

9-5 Working With Right Triangles

 When you complete the work for this section, you should be able to: Describe exactly how a right triangle is different from other kinds of triangles. Identify the right angle and hypotenuse of a right triangle. Write out and explain how to use the Pythagorean theorem. Given the lengths of three sides of a triangle, determine whether it is a right triangle. Given the lengths of two side of a right triangle, determine the length of the hypotenuse. Given the length of the hypotenuse and one side of a right triangle, determine the length of the remaining side.

You have seen that a triangle is any plane figure that has three sides. A triangle also has three angles. Get it? Tri- angle? A right triangle is a special kind of triangle. It is a right-angle triangle. One of the angles is an exact right angle — a 90-degree angle.

 Definition A right triangle is a triangle that includes one right, or 90-degree, angle. Introducing Right Triangles

Like any triangle, a right triangle has three sides. Two of the sides come together at the right angle. They are simply called sides. The third side is opposite the right angle. It is called the hypotenuse.

 Definition The side opposite the right angle of a right triangle is know as the hypotenuse. You can label the three sides and three angles in any way that is convenient for the problem at hand. However, there is a set of conventional names that are commonly used when discussing right triangles in a very general way.

 First, notice that the angles are labeled with upper-case letters A, B, and C. Angle C is always the right angle. Second, notice that the sides are labeled with lower-case letters a, b, and c. These sides are opposite the angles that have the corresponding upper-case letters:  side a is opposite angle A, side b is opposite angle B, and side c (the hypotenuse) is opposite angle C. The perimeter of a right triangle is equal to the sum of the lengths of the three sides. Using the conventional labels: P = a + b + c In words: Perimeter (P) is equal to side a plus side b plus side c.

The area of a right triangle is equal to one-half the product of side a and side b. Or in mathematical terms:

A = ½ab

Using the Pythagorean Theorem

For any right triangle, the lengths of the two sides and the hypotenuse are related by an important mathematical statement called the Pythagorean Theorem.

 Definition This formal statement of the Pythagorean Theorem relates the lengths of the three sides of a right triangle: c2 = a2 + b2 where c is the length of the hypotenuse, and a and b are the lengths of the two remaining sides.. Important:  The Pythagorean Theorem applies only to right triangles.

In words: The square of the hypotenuse of a right triangle is equal to the sum of the squares of the two sides.

Example 1

Try it out: Does this set of values obey the Pythagorean Theorem?  a = 20, b = 15, c = 25

 Substitute the given values into the Pythagorean Theorem c2 = a2 + b2 252 = 202 + 152 Square the terms 625 = 400 + 225 Complete the addition 625 = 625

Example 2 Identify the hypotenuse. The hypotenuse is located opposite the right angle. So the hypotenuse is the 20.c = 20 Identify the two remaining sides. The two remaining sides are 16 and 12. It makes no difference which you make side a and which is side b. So let:a = 16, b = 12 Substitute the values into the Pythagorean Theorem c2 = a2 + b2 202 = 162 + 122 Square the terms 400 = 256 + 144 Complete the addition 400 = 400

 Important The hypotenuse is always the longest of the three sides of a right triangle. The length of the hypotenuse is always less than the sum of the two other sides. If a triangle is a right triangle, the lengths of the three sides fit the Pythagorean Theorem

Example 3

Problem

The sides of a certain triangle measure 8 cm, 15 cm, and 17 cm.

1. If it is a right triangle, which measurement is the hypotenuse?
2. Is it really a right triangle?

Procedure

If it is indeed a right triangle, the hypotenuse would be the longest side:  17 cm.

If it is a right triangle, it will fit the Pythagorean Theorem:

c2 = a2 + b2
172
= 82 + 152
289 = 64 + 225
289 = 289 Solution

1. The hypotenuse is the 17 cm side.
2. Yes, it is a right  triangle.

Example 4

Problem

The sides of a certain triangle measure 15 in, 20 in, and 8 in.

1. If it is a right triangle, which measurement is the hypotenuse?
2. Is it really a right triangle?

Procedure

If it is indeed a right triangle, the hypotenuse would be the longest side:  20 in.

If it is a right triangle, it will fit the Pythagorean Theorem:

c2 = a2 + b2
202
= 152 + 82
400 = 225 + 64
400
¹ 289 Solution

1. If this were a right triangle, the hypotenuse would be the 20" side.
2. No, it is not a right triangle.

Exercise

 Work this exercise until you can accurately determine whether or not these dimensions can be the sides if a right triangle.

 Given the lengths of two sides, you can calculate the length of the third side ... even if you don't know the perimeter. If you are given the lengths of sides a and b, you can determine the length of side c — the hypotenuse. If you are given the length of the hypotenuse and one other side (a or b), you can determine the length to the remaining side. Equation

By solving the Pythagorean Theorem for c, you have an equation for determining the length of the hypotenuse: where:

c = the length of the hypotenuse of a right triangle
a, b = the lengths of the other two sides

Examples

Examples and Exercises

 Given the lengths of the two sides of a right triangle, determine the length of the hypotenuse.Round your answer to the nearest tenth, if necessary. Use these interactive examples and exercises to strengthen your understanding and build your skills.

Equation

By solving the Pythagorean Theorem for sides a and b, you have equations for determining the length of either side: and where:

c = the length of the hypotenuse of a right triangle
a, b = the lengths of the other two sides

Examples

Examples and Exercises

 Given the length of the hypotenuse and one side of a right triangle, find the length of the remaining side. Round your results to the nearest tenth, if necessary. Remember: The hypotenuse is the longest part of any right triangle. Use these interactive examples and exercises to strengthen your understanding and build your skills.

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