The Pythagorean Theorem One special kind of triangle is a right triangle a triangle with one interior angle of 90º. B A Note: In a polygon (like a triangle), the vertices (and corresponding angles) are labeled with capital letters, while the sides opposite each angle are labeled with lowercase letters. A In a right triangle, the longest side (the side opposite the 90º angle) is called the hypotenuse, while the other two sides of a right triangle are called the legs. One special relationship between the lengths of the sides of a right triangle is called the Pythagorean Theorem. It states that the sum of the squares of the measures of the legs equals the square of the measure of the hypotenuse. For a triangle labeled with a and b as the legs and c as the hypotenuse, the Pythagorean Theorem can be expressed: c b C B a C a 2 + b 2 = c 2 Pythagorean Theorem This means that if the measures of two of the sides of a right triangle are known, the measure of the third side can be found. example 1 The roads connecting the three towns on the map below form a right triangle. Two of the distances are given.
Based on the distances given on the map, what is the distance between Maple and Sable? A. 12 km B. 15 km C. 16 km D. 19 km In any problem dealing with a right triangle, always be sure to identify the hypotenuse (the longest side). In this problem, the longest side is the one between Maple and Hickory and has a length of 25 kilometers. This will be used as c in the Pythagorean Theorem, while the other given distance can be used as either a or b: The answer is B. example 2 The diagonal of a square is 25 units long. Which is the approximate length of a side of the square? A. 18 units B. 15 units C. 5 units D. 13 units A diagonal is a line segment that connect vertices of a polygon through the interior of the polygon, so in the case of a square, it s a line segment that runs diagonally through the square from one corner to the opposite corner: 25 diagonal
As can be seen in the previous diagram, a diagonal of a square divides that square into two right triangles, with the diagonal being the hypotenuse of both triangles: 25 25 Because both right triangles came from a square, the legs of each triangle have the same length: a 25 Now we can use the Pythagorean Theorem to find the approximate length of a: a The answer is A. example 3 The lengths of the legs of a right triangle are 5 cm and 10 cm. Which of the following measures is closest to the length of the hypotenuse? A. 11.2 cm B. 11.4 cm C. 11.6 cm D. 11.8 cm
Use the Pythagorean Theorem to solve this problem: The answer is A. Proving a Triangle to be a Right Triangle The Pythagorean Theorem can also be used to prove that a triangle is a right triangle. example 4 Is a triangle whose sides have lengths of 5, 12, and 13 a right triangle? To answer this question, first identify the length of the longest side: 13 in this example. Then use this value as c, the length of the hypotenuse, and substitute the other 2 values as a and b (it doesn t matter which one is a or b). These three lengths do make a right triangle. example 5 Is a triangle whose sides have lengths of 6, 7, and 9 a right triangle? This is not a right triangle.
example 6 A triangle is a right triangle if the lengths a, b, and c, of its three sides satisfy the following equation: Which of the following is a right triangle? A. a triangle with sides measuring 13, 15, and 19 B. a triangle with sides measuring 17, 25, and 36 C. a triangle with sides measuring 20, 21, and 29 D. a triangle with sides measuring 18, 23, and 41 To answer this question, just check the measures given in each answer and see if they work in the Pythagorean Theorem. The longest measure given must be used as c, but it doesn t matter which of the other two measures are chosen to be a and b: Answer A: Answer B: Answer C: Answer D: The answer is C.
Other Triangle Problems One thing to understand about any triangle, not just right triangles, is just common sense about how long the third side of a triangle could be given the measure of the other two sides. For example, the longest side of a triangle (and remember that the hypotenuse is the longest side of a right triangle only) can t be longer than the sum of the lengths of the other two sides. example 7 Eva has four sets of straws. The measurements of the straws are given below. Which set of straws could not be used to form a triangle? A. Set 1: 4 cm, 4 cm, 7 cm B. Set 2: 2 cm, 3 cm, 8 cm C. Set 3: 3 cm, 4 cm, 5 cm D. Set 4: 5 cm, 12 cm, 13 cm For any triangle, the longest side can t be longer than the other two sides combined. To find which answer can not be a triangle, find the longest measure of each set of straws and see if it is longer than the sum of the lengths of the other two straws: Answer A: The longest side is 7 cm, which is less than the sum of the other 2 sides (4 cm + 4 cm = 8 cm), so this set of straws could make a triangle. Answer B: The longest side is 8 cm, which is more than the sum of the other 2 sides (2 cm + 3 cm = 5 cm), so this set of straws could not make a triangle. Answer C: The longest side is 5 cm, which is less than the sum of the other 2 sides (3 cm + 4 cm = 7 cm), so this set of straws could make a triangle. Answer D: The longest side is 13 cm, which is less than the sum of the other 2 sides (5 cm + 12 cm = 17 cm), so this set of straws could make a triangle. The answer is B. (Note: Even though the correct answer was found after checking Answers A and B, it is a good idea to keep going and check all four answers just in case you made a mistake.)
Trigonometry Within a right triangle, there are three ratios involving the measures of two sides at a time that can be used for solving problems. B hypotenuse opposite side A adjacent side C In terms of A, the three sides of the right triangle can be labeled with reference to where that side is located relative to A. There is one side opposite A, a side next to (or adjacent to) A, and the hypotenuse. The three ratios are: These are usually abbreviated as: sin A = opp. hyp. cos A = adj. hyp. tan A = opp. adj. These ratios are called trigonometric functions (or trig. functions for short). example 8 Find the tan A, sin A, and cos A of the given triangle. A 5 4 B 3 C
On your calculator, either on a button or just above a button, you can see sin 1, cos 1, and tan 1. These are inverse functions, which will swap the angle and the value of the trig. function, as in the following example. From above, sin A = 5 3, so sin 1 3 = A 5 Using the inverse function on your calculator, you can find the measure of the angle itself: Note: When using these trig. functions, make sure your calculator is in degree mode and not radian mode.
Name If c is the measure of the hypotenuse of a right triangle and a and b are the legs of the triangle, find each missing measure. 1) a = 3, b = 4, c = 2) a = 5, b =, c = 13 3) a =, b = 1, c = 2 4) a = 2, b = 3, c = Determine whether the following lengths would form the sides of a right triangle. 5) 9, 40, 11 6) 13, 8, 9 7) 3, 2, 6 8) 20, 21, 29 Determine whether the given lengths could be the three sides of any triangle. 9) 5, 6, 9 10) 11, 14, 26 11) 11, 12, 23 12) 9, 15, 22
Use the figure below to answer question 13. A 5 4 B 3 C 13) Find sin A, cos A, and tan A. Use your calculator to solve question 14 16. (Make sure your calculator is in degree mode.) 14) If sin M = 0.7660, what is the measure of M? 15) If cos N = 0.3420, what is the measure of N? 16) If tan P = 1.804, what is the measure of P? For each triangle below, find the measure of the marked angle to the nearest degree. 17) 18) 10 3?? 9 4 For each triangle below, find the measure of the marked side. 19) 20)?? 3 30 12 45