Question: What is the radius of the incircle of the triangle whose sides measure 5, 12 and 13 units?

(A) 2 units
(B) 12 units
(C) 6.5 units
(D) 6 units
(E) 7.5 units

“What is the radius of the incircle of the triangle whose sides measure''- is a topic of the GMAT Quantitative reasoning section of the GMAT exam. GMAT quant section improves the analytical skills and mathematical proficiency of the candidates. It measures the candidates' abilities in solving highly calculative problems of mathematics. The students must know and enhance their learning regarding mathematical calculations to crack GMAT Problem Solving questions. The mathematical problems that represent the GMAT Quant topic in the problem-solving part can only be solved with better mathematical understanding. The candidates can practice questions by answering from the book “Kaplan GMAT Math Workbook”.

Solution and Explanation:

Approach Solution 1:

The problem statement informs that:

Given:

• A circle is inscribed within a triangle.
• The sides of the triangle are 5, 12 and 13 units.

Find Out:

• The radius of the incircle.

Since, 5, 12, and 13 are Pythagorean triplets, therefore, it is a right-angled triangle.
Let the radius of the incircle of the right-angled triangle is r.
As per the formula of the inradius of a right-angled triangle,
r = (a+b−c)/2 (where c is the hypotenuse, and a and b are the other two sides).
Therefore, by putting the value of the sides of the triangle, that is c= 13, a=5, b=12, we get,
r = (5+12-13)/2 = 2
Hence, the radius of the incircle= 2 units

Correct Answer: (A)

Approach Solution 2:

The problem statement suggests that:

Given:

• A circle is inscribed within a triangle.
• The sides of the triangle are 5, 12 and 13 units.

Find Out:

• The radius of the incircle.

Since, the sides of triangles 5, 12 and 13 represent the Pythagorean triplets, therefore, it is a right-angled triangle. Let’s analyse the question as per the diagram given below:

Here, since the two tangents pass from the same point on any circle are equal, then we can infer that, BP = BN = Radius r
Then, CN = CM = 12-r
Then, AP = AM = 5-r

Also from the question, we can say that AM + CM = 13
Or, in the next step, we can write it as (5-r) + (12-r) = 13
Therefore, r = 2.

Hence, the radius of the incircle= 2 units

Correct Answer: (A)

Approach Solution 3:

The problem statement declares that:

Given:

• A circle is inscribed within a triangle.
• The sides of the triangle are 5, 12 and 13 units.

Find Out:

• The radius of the incircle

Since 13^2 = 5^2 + 12^2, therefore it is a right-angled triangle, where the hypotenuse is 13 units, the base is 5 units and the height is 12 units.
As per the formula of the radius of the incircle,
Radius = 2* area of triangle/sum of sides
Therefore, area of triangle= ½* 5 * 12 = 30 (since area of triangle= ½ *base*height)
Therefore, the sum of sides= 5+12+13= 30
Hence radius of the incircle= 2* 30/30 = 2units

Correct Answer: (A)

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