# GMAT Practice Question - Right Triangles

Concept: Properties of right triangles. Equating area of a triangle using different methods.

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

#### Question: 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

Video explanation will be added soon

#### Equating areas using 2 different methods

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 = $\frac{\text{a + b + c}}{2} = \frac{5 + 12 + 13}{2} \\$ = 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.

#### 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 = $\frac{\text{a + b + c}}{2} = \frac{5 + 12 + 13}{2} \\$ = 15

Therefore, r = 15 - 13 = 2 units.

Choice A is the correct answer.

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