Example: mЄ1 = mЄ3 + mЄ5 mЄ4 = mЄ2 + mЄ5 mЄ6 = mЄ3 + mЄ2 ■ The sum of the exterior angles of a triangle equal to 360 degrees. Triangles More geometry questions on the GRE pertain to triangles than to any other topic. The following topics cover the information you will need to apply when solving triangle problems. CLASSIFYING TRIANGLES It is possible to classify triangles into three categories based on the number of equal sides: Scalene Triangle: no equal sides Isosceles Triangle: at least two equal sides Equilateral Triangle: all sides equal Scalene Isosceles Equilateral 3 4 6 5 1 2 – THE GRE QUANTITATIVE SECTION– 185 It is also possible to classify triangles into three categories based on the measure of the greatest angle: Acute Triangle: greatest angle is acute Right Triangle: greatest angle is 90 degrees Obtuse Triangle: greatest angle is obtuse ANGLE-SIDE RELATIONSHIPS Knowing the angle-side relationships in isosceles, equilateral, and right triangles will be useful when you take the GRE. ■ In isosceles triangles, equal angles are opposite equal sides. AB C m∠ A = m∠ B 50° 60° 70° 150° Obtuse Right Acute – THE GRE QUANTITATIVE SECTION– 186 ■ In equilateral triangles, all sides are equal and all angles are equal. ■ In a right triangle, the side opposite the right angle is called the hypotenuse. The hypotenuse is the longest side of the triangle. PYTHAGOREAN THEOREM The Pythagorean theorem is an important tool for working with right triangles. It states: a 2 + b 2 = c 2 , where a and b represent the length of the legs and c reprecents the length of the hypotenuse. This theorem allows you to find the length of any side as long as you know the measure of the other two. a 2 + b 2 = c 2 1 2 + 2 2 = c 2 1 + 4 = c 2 5 = c 2 ͙5 ෆ = c 2 1 √ ¯¯¯ 5 Hypotenuse Right Equilateral 60 6060 55 5 – THE GRE QUANTITATIVE SECTION– 187 45-45-90 RIGHT TRIANGLES A right triangle with two angles each measuring 45 degrees is called an isosceles right triangle. In an isosceles right triangle: ■ The length of the hypotenuse is ͙2 ෆ multiplied by the length of one of the legs of the triangle. ■ The length of each leg is multiplied by the length of the hypotenuse. x = ϫ ᎏ 1 1 0 ᎏ = = 5͙2 ෆ 30-60-90 R IGHT TRIANGLES In a right triangle with the other angles measuring 30 and 60 degrees: ■ The leg opposite the 30-degree angle is half the length of the hypotenuse. (And, therefore, the hypotenuse is two times the length of the leg opposite the 30-degree angle.) ■ The leg opposite the 60-degree angle is 3 times the length of the other leg. 10͙2 ෆ ᎏ 2 ͙2 ෆ ᎏ 2 10 x x ͙2 ෆ ᎏ 2 45° 45° – THE GRE QUANTITATIVE SECTION– 188 Example: x ϭ 2 ϫ 7 ϭ 14 and y ϭ ͙3 ෆ Circles A circle is a closed figure in which each point of the circle is the same distance from a fixed point called the center of the circle. A NGLES AND ARCS OF A CIRCLE ■ An arc is a curved section of a circle. A minor arc is smaller than a semicircle and a major arc is larger than a semicircle. M i n o r A r c M a j o r A r c Central Angle 60° 30° x y 7 60 30 2s s s Ί ෆ 3 – THE GRE QUANTITATIVE SECTION– 189 ■ A central angle of a circle is an angle that has its vertex at the center and that has sides that are radii. ■ Central angles have the same degree measure as the arc it forms. LENGTH OF ARC To find the length of an arc, multiply the circumference of the circle, 2πr,where r ϭ the radius of the circle, by the fraction ᎏ 36 x 0 ᎏ ,where x is the degree measure of the arc or central angle of the arc. Example: Find the length of the arc if x ϭ 36 and r ϭ 70. L = ᎏ 3 3 6 6 0 ᎏ × 2(π)70 L = ᎏ 1 1 0 ᎏ × 140π L = 14π A REA OF A SECTOR A sector of a circle is a region contained within the interior of a central angle and arc. A B C shaded region = sector r x r o – THE GRE QUANTITATIVE SECTION– 190 . with the other angles measuring 30 and 60 degrees: ■ The leg opposite the 30-degree angle is half the length of the hypotenuse. (And, therefore, the hypotenuse is two times the length of the leg. leg opposite the 30-degree angle.) ■ The leg opposite the 60-degree angle is 3 times the length of the other leg. 10 2 ෆ ᎏ 2 ͙2 ෆ ᎏ 2 10 x x ͙2 ෆ ᎏ 2 45° 45° – THE GRE QUANTITATIVE SECTION 188 Example: x. equal. ■ In a right triangle, the side opposite the right angle is called the hypotenuse. The hypotenuse is the longest side of the triangle. PYTHAGOREAN THEOREM The Pythagorean theorem is an important