# 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. #### GMAT Online CourseTry it free! Register in 2 easy steps and Start learning in 5 minutes! #### Already have an Account? #### GMAT Live Online Classes Next Batch Oct 12, 2021 #### GMAT Coaching in Chennai Next Batch after Lockdown #### GMAT Coaching in Chennai Sign up for a Free Demo of our GMAT Classes in Chennai ##### Where is Wizako located? Wizako - GMAT, GRE, SAT Prep An Ascent Education Initiative 14B/1 Dr Thirumurthy Nagar 1st Street Nungambakkam Chennai 600 034. India ##### How to reach Wizako? Phone:$91) 44 4500 8484
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