What Is The Area Of Triangle ABC ? GMAT Data Sufficiency

Question: What is the area of triangle ABC ?

(1) Side DC = 20
(2) Side AC = 8

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. EACH statement ALONE is sufficient.
5. Statements (1) and (2) TOGETHER are not sufficient.

“What is the area of triangle ABC?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Official Guide Quantative Review 2021". GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

Solution and Explanation

Approach Solution 1:

There is only one approach to solve the problem.

Let us look at the below image:

We need to find out what is the area of triangle ABC ?
The angle ACE is a straight line, hence it's 180°.
Now, since (angle BCD)=90°,
then (angle ACB) + (angle DCE)
= 180° - 90°
= 90°.
Next, in triangle ABC, (angle ACB) + (angle ABC) = 90°. Thus we have that:
(angle ACB) + (angle ABC) = 90°
= (angle ACB) + (angle DCE)
--> (angle ABC) = (angle DCE), which on the other hand implies that (angle ACB) = (angle CDE).
We can conclude that, all three angles in triangles ABC and CDE are equal, so these triangles are similar.
As per the formula, we know:
If two similar triangles have sides in the ratio x/y, then their areas are in the ratio \(x^2\)/ \(y^2\)
OR
in another way: in two similar triangles, the ratio of their areas is the square of the ratio of their sides:
→AREA / area = \(SIDE^2\)/\(side^2\)
(1) Side DC = 20
--> CE= \(\sqrt{20^2}\)-16^2
= 12→AC=20−12=8
-->\(AREA_{CDE}\)/ \(area_{ABC}\)
= \(16^2\)/ \(8^2\)
Since the area of triangle CDE = 16*12/2 = 96,
then 96 / \(area_{ABC}\)= \(16^2\)/ \(8^2\)
we can find the area od triangle ABC.
Hence, it is Sufficient.
(2) Side AC = 8. The same info as above.
This is also Sufficient.

Correct Answer: D

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