
top
Chapter 9—Basic Geometry 95 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 rightangle triangle. One of the angles is an exact right angle — a 90degree angle. Definition A right triangle is a triangle that includes one right, or 90degree, 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 uppercase letters A, B, and C. Angle C is always the right angle. Second, notice that the sides are labeled with lowercase letters a, b, and c. These sides are opposite the angles that have the corresponding uppercase 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 onehalf 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: c^{2} = a^{2} + b^{2} 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  c^{2} = a^{2} + b^{2 }25^{2} = 20^{2} + 15^{2}  Square the terms  625 = 400 + 225  Complete the addition  625 = 625  Example 2 How about this triangle? Do the given lengths of the sides follow the Pythagorean Theorem? 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  c^{2} = a^{2} + b^{2 }20^{2} = 16^{2} + 12^{2}  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.  If it is a right triangle, which measurement is the hypotenuse?
 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: c^{2} = a^{2} + b^{2 }17^{2} = 8^{2} + 15^{2} 289 = 64 + 225 289 = 289 Solution  The hypotenuse is the 17 cm side.
 Yes, it is a right triangle.
Example 4 Problem The sides of a certain triangle measure 15 in, 20 in, and 8 in.  If it is a right triangle, which measurement is the hypotenuse?
 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: c^{2} = a^{2} + b^{2 }20^{2} = 15^{2} + 8^{2} 400 = 225 + 64 400 ¹ 289 Solution  If this were a right triangle, the hypotenuse would be the 20" side.
 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.  
