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COMPLETE GRE GUIDE 2015 Know and Use Facts about triangles By remembering these facts about triangles, you can often save yourself a lot of time and trouble. Math StratEg y18 I. a x° y° b If a 5 b, then x 5 y The base angles of an isosceles triangle are equal. a x° y° b If x 5 y, then a 5 b If the base angles of a triangle are equal, the triangle is isosceles. II. z°x° y° A B D C 1 2 A B D C 1 2 is a straight line. Then, x 5 y 1 z The measure of an exterior angle is equal to the sum of the measures of the remote interior angles. III. a x° y° b If a , b, then y , x a x° y° b If y , x, then a , b In a triangle, the greater angle lies opposite the greater side. IV. Similar Triangles A B C a bc f e d D E F If \u394ABC ~ \u394DEF, then m+A 5 m+D m+B 5 m+E m+C 5 m+F m A m D m B m E m C m F d a e b f cand = = = = = B B B B B B V. A B C m+A 1 m+B 1 m+C 5 180º The sum of the interior angles of a triangle is 180 degrees. VI. A B D C ABC AD BC2Area of 3 #= The area of a triangle is one-half the product of the altitude to a side and the side. Note: If m +A 5 90°, Area also Area also AB AC2 #= GRE2015_P04.indd 118 4/18/14 11:29 AM STRATEGY SECTION \u2022 119 VII. a c b x° y° In a right triangle, c2 5 a2 1 b2 and x°1 y° 5 90° VIII. Memorize the following standard triangles: 15 17 8 7 5 12 24 54 2 1 1 1 1 1 1 1 60° 60° 60° 60° 45° 45° 45° 45° 30° 3 9 4140 13 25 3 2 2 2 2 2 IX. ba c a 1 b . c a 1 c . b b 1 c . a The sum of the lengths of two sides of a triangle is greater than the length of the third side. (This is like saying that the shortest distance between two points is a straight line.) example 1 In the diagram below, what is the value of x? 24 x 10 M N P (A) 20 (B) 25 (C) 26 (D) 45 (E) 48 Choice C is correct. Method 1: Use Statement VII. Then, x2 5 242 1 102 5 576 1 100 5 676 Thus, x 5 26 (Answer) Method 2: Look at Statement VIII. Notice that \u394MNP is similar to one of the standard triangles: 24 x 10 M N P 13 12 5 This is true because 24 12 10 5 = (Look at IV). = =, 26 ( )x or x Answer24 12 13Hence example 2 If Masonville is 50 kilometers due north of Adamston and Elvira is 120 kilometers due east of Adamston, then the minimum distance between Masonville and Elvira is (A) 125 kilometers (B) 130 kilometers (C) 145 kilometers (D) 160 kilometers (E) 170 kilometers Choice B is correct. Draw a diagram first. Masonville 50 km 120 kmAdamston Elvira x The given information translates into the diagram above. Note Statement VIII. The triangle above is a multiple of the special 5\u201312\u201313 right triangle. GRE2015_P04.indd 119 4/18/14 11:29 AM 120 \u2022 GRUBER\u2019S COMPLETE GRE GUIDE 2015 50 5 10(5) 120 5 10(12) Thus, x 5 10(13) 5 130 kilometers (Note: The Pythagorean Theorem could also have been used: 502 1 1202 5 x2.) example 3 C b° a° c° B A (Note: Figure is not drawn to scale.) In triangle ABC, if a . c, which of the following is true? (A) BC 5 AC (B) AB . BC (C) AC . AB (D) BC . AB (E) BC . AC Choice D is correct. (Remember triangle inequality facts.) From basic geometry, Statement III, we know that, since m+BAC . m+BCA, then leg opposite +BAC . leg opposite +BCA, or BC . AB example 4 C 45° B A (Note: Figure is not drawn to scale.) The triangle above has side BC 5 10, angle B 5 45°, and angle A 5 90°. The area of the triangle (A) is 15 (B) is 20 (C) is 25 (D) is 30 (E) Cannot be determined. Choice C is correct. First find angle C using Statement V. 90° 1 45° 1 m+C 5 180° So m+C 5 45°. Using Statement I, we find AB 5 AC, since m+B 5 m+C 5 45°. Since our right triangle ABC has BC 5 10, using Statement VIII (the right triangle , ,2 2 2 2 1), multiply by 10 to get a right triangle: , ,2 10 2 2 10 2 10 5AB 2 10 2 2Thus side = = 5AC 2 10 2 2side = = Now the area of triangle ABC, according to Statement VI, is 5 5 252 2 2 2 25 2# #= = example 5 C x° x°55° 45° D B A In the figure above, what is the value of x? (A) 30 (B) 40 (C) 50 (D) 80 (E) 100 Choice B is correct. Remember triangle facts. Use Statement II. \u2220ADB is an exterior angle of \u394 ACD, so m\u2220ADB 5 x 1 x 5 2x 1 In \u394ADB, the sum of its angles 5 180 (Statement V), so m\u2220ADB 1 55 1 45 5 180 or m\u2220ADB 1 100 5 180 or m\u2220ADB 5 80 2 Equating 1 and 2 , we have 2x 5 80 x 5 40 (Answer) GRE2015_P04.indd 120 4/18/14 11:29 AM STRATEGY SECTION \u2022 121 When Calculating answers, Never Multiply and/or Do Long Division If you Can reduce First Note: On the GRE exam, because calculators are permitted, you may do the following problems with a calculator also. But it would be wise for you to see the other approach too\u2014how the problem can be solved without the use of a calculator. Math StratEg y19 example 1 If , thenw w45 40 81 150 # #= = (A) 3 (B) 6 4 3 (C) 7 4 1 (D) 9 (E) 20 4 1 Do not multiply in} 81 3 150 and 45 3 40 to getthis case. , , 1 800 12 150 Factor first: 9 5 4 10 9 9 45 40 81 150 15 10Factor first # # # # # # ? ? S S Then cancel like factors in numerator and denominator: 9 5 4 10 9 9 15 10 # # # # # # Reduce further: 5 9 5 4 3 Reduce further # # # Then simplify: 6 ( )Answer4 27 4 3 = Thus, Choice B is correct. example 2 3 3 3 4 4 4 3 3 3 2 2 2 + + + + = (A) 27 16 (B) 9 8 (C) 3 4 (D) 27 64 (E) 81 512 Choice A is correct. example 6 5 4 a (Note: Figure is not drawn to scale.) Which of the following represent all of the possibilities for the value of a in the figure above? (A) 1 , a , 9 (B) 4 , a , 5 (C) 0 , a , 9 (D) 4 , a , 9 (E) 5 , a , 9 Choice A is correct. From Statement IX, since the sum of the lengths of two sides of a triangle is greater than the length of the third side, we have: a 1 5 . 4 1 a 1 4 . 5 2 5 1 4 . a 3 From 2 we get: a . 1. From 3 we get: 9 . a. This means that 9 . a . 1, or 1 , a , 9. GRE2015_P04.indd 121 4/18/14 11:29 AM 122 \u2022 GRUBER\u2019S COMPLETE GRE GUIDE 2015 3 3 3 4 4 4 3 3 3 2 2 2 + + + + = Factor and reduce: 3( ) 3( ) 3 4 3 2 5 27 16 (Answer) example 3 6 7 8 9 12 14 ,x x 18If then# # # # #= = (A) 2 1 (B) 1 (C) 4 (D) 8 (E) 12 Choice B is correct. Given: 6 7 8 9 12 14x 18 # # # # #= 1 6 7 8 12 14x 9 18so that # # # # #= 2 Do not multiply the numbers out in the numerator and denominator of 2 ! It is too much work! Rewrite 2 . Factor and reduce: x 5 6 7 8 12 14 6 7 8 9 2 6 2 7 2 9 2 2 8 8 1 ( )Answer 9 18 8 2 # # # # # # # # # # # # # # # = = = = 6 7 8 12 14 6 7 8 9 2 6 2 7 2 9 2 2 8 8 1 ( )Answer 9 18 8 2 # # # # # # # # # # # # # # # = = = = (Answer) example 4 If 21,y y27 81 then# = = (A) 21 1 (B) 7 1 (C) 3 (D) 7 (E) 21 Choice D is correct. : 81 21Given y27 # = Multiply both sides by 27 to get 81 3 y 5 21 3 27. y 81 21 27#= Factor and reduce: 9 9 3 7 3 9 3 3 3 3 7 y y 7 $ $ # $ $ $ # = = = (Answer) example 5 Find the value of 2 7 10 2y y y22 + rounded to the nearest whole number if y 5 8.000001. (A) 2 (B) 3 (C) 5 (D) 6 (E) 16 Choice B is correct. Given: 2 7 10