Question: If the lengths of the legs of a right triangle are integers, what is the area of the triangular region?

(1) The length of one leg is 3/4 the length of the other.
(2) The length of the hypotenuse is 5.

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.

Correct Answer: B

Solution and Explanation:
Approach Solution 1:
The problem statement states that:
Given:

• The lengths of the legs of a right triangle are integers.

Find out:

• The area of the triangular region.

This information is evidently huge since there are no multiple right triangles that hold integer lengths.
The sets of 3 integers that could be the lengths of right triangles are known as Pythagorean Triplets.
A few such triplets are as follows:
3-4-5, 5-12-13, 7-24-25, 8-15-17, etc
It is necessary to note that we can make additional triplets by multiplying each of the above triplets by some integer value.
For example, since 3-4-5 is a Pythagorean triplet, we also know that the following multiples will also be triplets: 6-8-10, 9-12-15, 12-16-20, 21-28-35, ....etc.

Statement 1 alone: The length of one leg is 3/4 the length of the other.
There are infinitely many right triangles that fulfil this condition. Here are two:
Case 1: The triangle has lengths 3, 4 and 5.
In this case, the area of the triangular region = (3)(4)/2 = 6
Case 2: The triangle has lengths 6, 8 and 10.
In this case, the area of the triangular region = (6)(8)/2 = 24
Therefore, we cannot find the area of the triangular region with certainty.
Hence, Statement 1 is NOT SUFFICIENT.

Statement 2 alone: The length of the hypotenuse is 5
In each Pythagorean triplet, the greatest value always symbolises the length of the hypotenuse.
When we go through all of the possible Pythagorean triples, there is only one triplet whose greatest value is 5.
Therefore, statement 2 illustrates that the triangle must be a 3-4-5 right triangle.
Hence, the area of the triangle = (3)(4)/2 =6
Thus, from this statement, we can find the area of the triangular region with certainty.
Hence, statement 2 alone is SUFFICIENT

Approach Solution 2:
The problem statement informs that:
Given:

• The lengths of the legs of a right triangle are integers.

Find out:

• The area of the triangular region.

The triangle is a right triangle
Therefore, to solve the question we can follow the rules of Pythagorean's theorem:
a^2 + b^2 = c^2

Statement 1: The length of one leg is 3/4 the length of the other.
There are an infinite number of Pythagorean triples that have one leg equal to 3/4 of the other.
Side lengths could be: 3,4,5...... 6,8,10......9,12,15......12,16,20......etc
Hence, statement 1 is Not Sufficient
Statement 2: The length of the hypotenuse is 5.
The only right triangle that has a hypotenuse of 5, with all sides being integers, is the 3,4,5 Pythagorean triple.
Since from this statement, we can find all the sides of the triangle, therefore, we can derive the area of the triangle.
Hence, statement 2 is Sufficient to find the area of the triangular region.

Approach Solution 3:
The problem statement implies that:
Given:

• The lengths of the legs of a right triangle are integers.

Find out:

• The area of the triangular region.

Statement 1: The length of one leg is 3/4 the length of the other.
We can derive many values that satisfy this scenario.
Hence, statement 1 is Not Sufficient to find the area of a triangle.

Statement 2: The length of the hypotenuse is 5.
Since we are said that the lengths of the legs are integers, these legs must be 3 and 4.
Thus we can find the area of the triangle.
Hence, statement 2 is SUFFICIENT.

“If the lengths of the legs of a right triangle are integers”- is a topic of the GMAT Quantitative reasoning section of the GMAT exam. This question has been taken from the book “GMAT Official Guide 2021”. The GMAT Quant section comprises a total of 31 questions. GMAT Data Sufficiency questions come up with a problem statement followed by two factual statements. GMAT data sufficiency includes 15 questions which are two-fifths of the total 31 GMAT quant questions.

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