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GRE Practice - Quantitative Reasoning - Set 25

QR25-1
  1. The figure above shows the dimensions of an isosceles triangle in terms of x. What is the area of the triangle?

2x+3 = 3x-2 [sides opposite to equal angles]
x = 5
Equal sides = 13
Height: 12
Using Pythagoras theorem we get, base of each half-triangle = 5
Using the property that the perpendicular to the base of an isosceles triangle bisects the base: , we get the base = 10
Area = ½ x Base x Height = ½ x 10 x 12 = 60

  1. If the perimeter of an isosceles triangle is 40, then which of the following CANNOT be the length of the unequal side?
Let the 2 equal sides be x each and the unequal side be y Perimeter: 2x + y = 40 Now, in any triangle the sum of two sides must be greater than the third side So x+y >y and x+x > y Or the sum of the equal sides is greater than the unequal side. Which means the sum of the equal sides is greater than 40/2 = 20 In other words, y < 20
QR25-3

In the figure above, If ABCD is a square and AE = EB, then which of the following statements is NOT true?
Select all that apply.

  1. DE = CE
  2. p = t
  3. q = s
  4. r = s
  5. r = 30

Using Pythagoras theorem:
In ∆AED AE2 + AD2 = DE2
In ∆BEC BE2 + BC2 = CE2
But AD = BC [sides of a square] and AE = BE [Given]
So DE = CE A is TRUE
DE = CE so the angles opposite must also be equal p = t B is TRUE
q=90-p and s = 90 – t so q = s C is TRUE
r = s if BE = BC but BE = ½ BC. D is FALSE
We know in a 30-60-90 triangle, if r = 30 then BE = √3BC but BE = ½ BC. E is FALSE.

  1. If √x = 3 and √y = -2, then what is the value of x2 + y2 +2xy in terms of y?

√x = 3 so x = 9
√y = -2 so y = 4
x2 + y2 +2xy = (x+y)2 = 132 = 169

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