McGraw-Hill Education GRE 2019

6. C Recall that the diagonal of a square splits the square into two right
   triangles. Since leg BC of 45-45-90 triangle ABE = the side of BCDE, the
   hypotenuse of ABE = the diagonal of BCDE. Thus ABE =^12 of BCDE. The two
   quantities are equal.


7. B The shared side of the two triangles is AC. The perimeter of triangle
   ACD = 3(AC). Since AC is the hypotenuse of a right triangle, the legs must be
   less than AC. Thus the perimeter of ABC will be less than 3(AC) and therefore
   less than the perimeter of ACD.


8. B The area of XYQ =^12 × XQ × YQ. The area of YQZ =^12 × QZ × YQ. Thus
   1
   2 × XQ × YQ <

1

2 × QZ × YQ^ →^ XQ < QZ

To solve for XY, use the Pythagorean theorem: (XQ)^2 + (YQ)^2 = (XY)^2. To

solve for YZ, use the Pythagorean theorem: (QZ)^2 + (YQ)^2 = (YZ)^2. Since both

triangles share side YQ and XQ < QZ, XY must be less than YZ.

9. C Since AB = BC, angle BAC = angle BCA.
   BAC + BCA = 110
   BAC = 55 and BCA = 55
   If ∠BAC = 55, then ∠BAD = 55 – 30 = 25. Since AD = DC, angle ∠DCA = 30.
   If ∠DCA = 30 and ∠BCA = 55, then ∠BCD = 55 – 30 = 25. The two quantities
   are equal.

Polygons

A polygon is any two-dimensional shape. Familiar examples of polygons are

squares, triangles, and rectangles. Here are some definitions that you will find

useful in working with polygons:

■ The Sum of the Angles: The sum of the angles of any polygon can be

determined by using the following formula: sum of the angles = (n – 2)180,

where n = the number of sides.

■ The Perimeter of a Polygon: The perimeter of any polygon will be the sum of

its side lengths.

■ Quadrilaterals: A quadrilateral is any four-sided polygon. The sum of the

angles of any quadrilateral is 360. The quadrilaterals most commonly tested

on the GRE are parallelograms.

■ Parallelograms: A parallelogram is a quadrilateral in which opposite sides

are parallel.

A D

B C

Parallelogram

CHAPTER 13 ■ GEOMETRY 397

04-GRE-Test-2018_313-462.indd 397 12/05/17 12:04 pm