ABCD Is An Isosceles Trapezium GMAT Problem Solving

Question - ABCD is an isosceles trapezium. If EG||DC, AE||BF, FH||GC, BG = GC = HC, and ∠AEG=60°∠AEG=60°, what is the area of the shaded region?

1. 25√3
2. 30√3
3. 35√3
4. 40√3
5. 50√3

ABCD is an isosceles trapezium.’- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

Solution and Explanation:

Approach Solution 1

It is given ABCD is an isosceles trapezium. If EG||DC, AE||BF, FH||GC, BG = GC = HC, and ∠AEG=60°∠AEG=60°, what is the area of the shaded region?

4 equilateral triangles, each with a side length of 5, are used to calculate the area of the shaded region.
Total area = 4 * [\(\frac{\sqrt{3+5^2}}{4}\)]
25\(\sqrt{3}\)
The answer is A which is 25\(\sqrt{3}\)

Correct Answer: A

Approach Solution 2

There is another approach to answering this question which is Pretty simple
Join AF and triangle. BF = BG = 5 since BFG is an equilateral triangle. Since we also know that AB = 5, we can conclude that triangle ABF is an equilateral triangle. As a result, the upper half has three equilateral triangles, while the lower portion has five. Three equilateral triangles would make up the bottom shaded region, but only one equilateral triangle exists in the top shaded region.
Total area = 4 * [\(\frac{\sqrt{3+5^2}}{4}\)]
25\(\sqrt{3}\)
The answer is A which is 25\(\sqrt{3}\)

Approach Solution 3

Connecting the dots with AF, triangle BFG is an equilateral triangle so BF = BG = 5. As it mentioned in the problem AB = 5 too so we can find triangle ABF is an equilateral triangle too. Hence, the top portion consists of 3 equilateral triangles, the bottom portion would consist of 5 equilateral triangles. The bottom shaded region must be 3 equilateral triangles. The top shaded region is one equilateral triangle.

The shaded area would be 4∗3√4∗5^2=25\(\sqrt{3}\)

Correct Answer: A

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