GMAT Quant Questions | Geometry #5

Properties of right triangles | Incircle of triangles

This GMAT quant practice question is a problem solving question in Geometry. Concept: Finding the radius of the circle inscribed in a triangle.

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

  1. 2 units
  2. 12 units
  3. 6.5 units
  4. 6 units
  5. 7.5 units

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Explanatory Answer | GMAT Geometry

Compute area of the right triangle using two methods and equate the two

5, 12, and 13 is a Pythagorean triplet. So, the triangle is a right triangle.
Area using Method 1: Area of a triangle = \\frac{1}{2}) × b × h, where 'b' is the base and 'h' is the height of the triangle.
In this right triangle, if the base is 12, the height will be 5 or vice versa.
In either case, area = \\frac{1}{2}) × 5 × 12 = 30 sq units.

Area using Method 2: Area of a triangle = r × s, where 'r' is the radius of the inscribed circle and 's' is the semi perimeter of the triangle.
Semi perimeter = \\frac{a + b + c}{2}) = \\frac{5 + 12 + 13}{2}) = 15
Equating the area found using the First method to that using the second one, 30 = r × 15
Or r = 2 units.

Choice A is the correct answer.


Alternative Method: Formula based on equating tangents to a circle from an external point

Note: In a right angled triangle, the radius of the incircle = s - h,
where 's' is the semi perimeter of the triangle and 'r' is the radius of the inscribed circle.
The semi perimeter of the triangle = \\frac{a + b + c}{2}) = \\frac{5 + 12 + 13}{2}) = 15
Therefore, r = 15 - 13 = 2 units.
Choice A is the correct answer.



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