Types of Triangles You can classify triangles into three categories based on the number of equal sides. ■ Scalene Triangle: no equal sides ■ Isosceles Triangle: two equal sides ■ Equilateral Triangle: all equal sides You also can classify triangles into three categories based on the measure of the greatest angle: ■ Acute Triangle: greatest angle is acute 50° 60° 70° Acute Equilateral Isosceles Scalene –GEOMETRY REVIEW– 109 ■ Right Triangle: greatest angle is 90° ■ Obtuse Triangle: greatest angle is obtuse Angle-Side Relationships Understanding the angle-side relationships in isosceles, equilateral, and right triangles is essential in solving ques- tions on the SAT. ■ In isosceles triangles, equal angles are opposite equal sides. ■ In equilateral triangles, all sides are equal and all angles are 60°. 60º 60º60º ss s 22 m∠a = m∠b 130° Obtuse Right –GEOMETRY REVIEW– 110 ■ In right triangles, the side opposite the right angle is called the hypotenuse. Practice Question Which of the following best describes the triangle above? a. scalene and obtuse b. scalene and acute c. isosceles and right d. isosceles and obtuse e. isosceles and acute Answer d. The triangle has an angle greater than 90°, which makes it obtuse. Also, the triangle has two equal sides, which makes it isosceles. Pythagorean Theorem The Pythagorean theorem is an important tool for working with right triangles. It states: a 2 ϩ b 2 ϭ c 2 ,where a and b represent the lengths of the legs and c represents the length of the hypotenuse of a right triangle. Therefore, if you know the lengths of two sides of a right triangle, you can use the Pythagorean theorem to determine the length of the third side. 40° 100° 40° 66 Hypotenu se –GEOMETRY REVIEW– 111 Example a 2 ϩ b 2 ϭ c 2 3 2 ϩ 4 2 ϭ c 2 9 ϩ 16 ϭ c 2 25 ϭ c 2 ͙25 ෆ ϭ ͙c 2 ෆ 5 ϭ c Example a 2 ϩ b 2 ϭ c 2 a 2 ϩ 6 2 ϭ 12 2 a 2 ϩ 36 ϭ 144 a 2 ϩ 36 Ϫ 36 ϭ 144 Ϫ 36 a 2 ϭ 108 ͙a 2 ෆ ϭ ͙108 ෆ a ϭ ͙108 ෆ a 12 6 4 c 3 –GEOMETRY REVIEW– 112 Practice Question What is the length of the hypotenuse in the triangle above? a. ͙11 ෆ b. 8 c. ͙65 ෆ d. 11 e. 65 Answer c. Use the Pythagorean theorem: a 2 ϩ b 2 ϭ c 2 ,where a ϭ 7 and b ϭ 4. a 2 ϩ b 2 ϭ c 2 7 2 ϩ 4 2 ϭ c 2 49 ϩ 16 ϭ c 2 65 ϭ c 2 ͙65 ෆ ϭ ͙c 2 ෆ ͙65 ෆ ϭ c Pythagorean Triples A Pythagorean triple is a set of three positive integers that satisfies the Pythagorean theorem, a 2 ϩ b 2 ϭ c 2 . Example The set 3:4:5 is a Pythagorean triple because: 3 2 ϩ 4 2 ϭ 5 2 9 ϩ 16 ϭ 25 25 ϭ 25 Multiples of Pythagorean triples are also Pythagorean triples. Example Because set 3:4:5 is a Pythagorean triple, 6:8:10 is also a Pythagorean triple: 6 2 ϩ 8 2 ϭ 10 2 36 ϩ 64 ϭ 100 100 ϭ 100 7 4 –GEOMETRY REVIEW– 113 Pythagorean triples are important because they help you identify right triangles and identify the lengths of the sides of right triangles. Example What is the measure of ∠a in the triangle below? Because this triangle shows a Pythagorean triple (3:4:5), you know it is a right triangle. Therefore, ∠a must measure 90°. Example A right triangle has a leg of 8 and a hypotenuse of 10. What is the length of the other leg? Because this triangle is a right triangle, you know its measurements obey the Pythagorean theorem. You could plug 8 and 10 into the formula and solve for the missing leg, but you don’t have to. The triangle shows two parts of a Pythagorean triple (?:8:10), so you know that the missing leg must complete the triple. Therefore, the sec- ond leg has a length of 6. It is useful to memorize a few of the smallest Pythagorean triples: 3:4:5 3 2 + 4 2 = 5 2 6:8:10 6 2 + 8 2 = 10 2 5:12:13 5 2 + 12 2 = 13 2 7:24:25 7 2 + 24 2 = 25 2 8:15:17 8 2 + 15 2 = 17 2 810 ? 35 a 4 –GEOMETRY REVIEW– 114 . side. 40 ° 100° 40 ° 66 Hypotenu se –GEOMETRY REVIEW 111 Example a 2 ϩ b 2 ϭ c 2 3 2 ϩ 4 2 ϭ c 2 9 ϩ 16 ϭ c 2 25 ϭ c 2 ͙25 ෆ ϭ ͙c 2 ෆ 5 ϭ c Example a 2 ϩ b 2 ϭ c 2 a 2 ϩ 6 2 ϭ 12 2 a 2 ϩ 36 ϭ 144 a 2 ϩ 36 Ϫ 36 ϭ 144 . Pythagorean triples: 3 :4: 5 3 2 + 4 2 = 5 2 6:8:10 6 2 + 8 2 = 10 2 5:12:13 5 2 + 12 2 = 13 2 7: 24: 25 7 2 + 24 2 = 25 2 8:15:17 8 2 + 15 2 = 17 2 810 ? 35 a 4 –GEOMETRY REVIEW 1 14 . 36 ϭ 144 Ϫ 36 a 2 ϭ 108 ͙a 2 ෆ ϭ ͙108 ෆ a ϭ ͙108 ෆ a 12 6 4 c 3 –GEOMETRY REVIEW 112 Practice Question What is the length of the hypotenuse in the triangle above? a. ͙11 ෆ b. 8 c. ͙65 ෆ d. 11 e.