Question: In the figure shown below, the area of square region ACEG is 729, and the ratio of the area of square region IDEF to the area of square region ABHI is 1 to 4. What is the length of segment CD?

(A) 12
(B) 15
(C) 18
(D) 24
(E) 36

Correct Answer: (C)

Solution and Explanation:
Approach Solution 1:

The problem statement informs that:

Given:

• The area of square region ACEG is 729
• The ratio of the area of square IDEF to the area of square ABHI is 1 to 4.

Find Out:

• The length of segment CD

Let the length of AH = a and HG =b

Since ABHI is a square, AH =AB = a.
In the same way, ID = IF =HG = b since IDEF is a square.

Therefore, the length of one side of square ACEG = a +b
(a+b)^2 = 729, since the area of square ACEG is 729
Since the ratio of the area of square region IDEF to the area of square region ABHI is 1 to 4.
Area of square region IDEF = b^2
Area of square region ABHI = a^2
Therefore, b^2: a^2= 1/4 = (1/2)^2
Therefore we can say, b/a = ½ or -½ ( the ratio of 2 positive numbers can not be negative, hence rejected)
b/a = ½
a = 2b

Therefore, by putting the value of a in the equation, we get,
(a+b)^2 = 729
=>(3b)^2 = 729
=>9b^2 = 729
=>b^2 = 81
=>b = 9
=>a = 18.

Therefore, the length of CD = a = 18

Approach Solution 2:

The problem statement suggests that:

Given:

• The area of square region ACEG is 729
• The ratio of the area of square IDEF to the area of square ABHI is 1 to 4.

Find Out:

• The length of segment CD

Area of square ACEG = 729
Therefore, we can say, CE^2 = 729 (since area of square = Side^2)
Therefore, CE = 27 . . . (i)
The ratio of the area of square region IDEF to the area of square region ABHI = 1 : 4
Therefore, we can say, DE^2 : BI^2 = 1 : 4
Hence, DE : BI = 1 : 2

Since CDIB is a rectangle, then CD = BI
Therefore, we can say, DE : CD = 1 : 2 . . . (ii)

Hence, from equations (i) and (ii), we get:
CD = (2/ 1 + 2) × 27 = 18
Therefore, the length of CD = 18

“In the figure shown below, the area of square region ACEG is 729''- is a topic of the GMAT Quantitative reasoning section of the GMAT exam. GMAT Problem Solving questions test candidates’ content and numerical literacy to solve mathematical problems. GMAT Quant practice papers cite several sorts of questions that will help the candidates to enhance their mathematical knowledge.

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