What is the Circumradius of a Right Angle Triangle if its Sides are GMAT Data Sufficiency

Question: What is the circumradius of a right-angle triangle if its sides are integer numbers?

i) If one of the sides is 6 and hypotenuse is 10.
ii) If one of the sides is 6 and the area is 24.

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: D

Solution and Explanation:
Approach Solution 1:
This question can be solved using only one approach solution.

Given to us in the question that the sides of a right-angled triangle are integer numbers. It is asked to find out the circumradius of the right-angled triangle.
There are two statements given, we have to check whether these statements are sufficient to get the answer or not.
The statements are –
i) If one of the sides is 6 and hypotenuse is 10.
ii) If one of the sides is 6 and the area is 24.
This is a question from coordinate geometry.

It is important for the candidate to know the formula of the circumradius of a right-angled triangle.
The formula for the circumradius is
R = 2s - r
But when a triangle is a right-angled triangle the circumradius is half of the hypotenuse.
In statement 1 it is given that one of the sides is 6 and hypotenuse is 10
The circumradius will be hypotenuse /2 = 10/2 = 5
Therefore this statement is sufficient to find the answer.
In second statement, it is given that one side is 6 and the area is 24
We have to find the hypotenuse.
Area = \(\frac{1}{2}\)* b* h = 24
= \(\frac{1}{2}\) * 6 * h = 24
=> h = 24 *2/6
=> h = 8
Let l be the hypotenuse
L =\(\sqrt{h^2+b^2}\)= \(\sqrt{8^2+6^2}\) = 100 = 10
The hypotenuse is 10 units
The circumradius of the triangle will be half of hypotenuse.
R = 10/ 2 = 5 units
This statement is also sufficient to get the answer.

“What is the circumradius of a right-angle triangle if its sides are”- is a topic of the GMAT Quantitative reasoning section of GMAT. 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.

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