Question

In the figure above, ABCD, BEFG, and DHIJ are squares with AB= 2(DH) and DH = 2(BE). If a point is chosen at random inside square ABCD, what is the probability it will be in the shaded region?

(A) 1/2
(B) 5/8
(C) 11/16
(D) 3/4​

Answer 1

`(11)/(16)\\\\`
Option C

Explanation:

We have the following information:

AB = 2DH

DH = 2BE or BE = DH/2

The area of square ABCD = (2DH)² = 4DH²

The area of square BEFG = BE² = (DH/2)² = DH²/4

The area of square DHFJ = DH²

Area of the unshaded region

= area of square BEFG + area or square DHFJ

= DH²/4 + DH² = DH²/4 + 4DH²/4 = 5DH²/4

Area of shaded region

= area of square ABCD - area of unshaded region

`= AB^2 - (5)/(4)DH^2\\\\= 4DH^2 - (5)/(4) DH^2\\\\= (4DH^2 \cdot 4)/(4) - (5)/(4) \cdot DH^2`

`= (16DH^2 - 5DH^2)/(4)= (11DH^2)/(4)`

`\textrm{Probability of a point falling in the shaded region} =\frac{\textrm{Area of shaded region}}{\textrm{Total area of both regions}}\\\\= (11DH^2/4)/(4DH^2) = (11)/(16)\\\\`