Bunuel wrote:
What is the radius of the inscribed circle to a triangle whose sides measure 21cm, 72cm and 75cm respectively?
(A) 37.5 cm
(B) 28.5 cm
(C) 14.5 cm
(D) 12.5 cm
(E) 9 cm
Solution:In general, it’s difficult to find the radius of a circle inscribed in a triangle if you don’t know any special formulas to find it. However, there is a special formula which states that the radius of a circle inscribed in a triangle is the quotient of the area of the triangle and its semiperimeter. That is, r = A/s, where r is the radius of the inscribed circle and A and s are the area and semiperimeter of the circumscribed triangle, respectively. Notice that if a, b, and c are the sides of the triangle, then the semiperimeter s is equal to (a + b + c)/2.
Here, we see that the circumscribed triangle is a right triangle since 21^2 + 72^2 = 441 + 5184 = 5625 = 75^2. Therefore, its area is (21 x 72)/2 = 756. The semiperimeter is (21 + 72 + 75)/2 = 84. Therefore, the radius of the inscribed circle is 756/84 = 9 cm.
Alternate Solution:Since 21 = 3 x 7, 72 = 3 x 24, and 75 = 3 x 25; the triangle is a 7-24-25 right triangle.
Let the radius of the inscribed circle be r and draw the radii connecting the center of the circle to the edges of the triangle. Since the edges of the triangle are tangent to the circle, each radius will meet each edge of the triangle at a right angle. Next, connect the three vertices of the triangle to the center of the circle. By doing this, we divided the triangle into three smaller triangles. The bases of these triangles have lengths of 21, 72, and 75; and in each triangle, the height corresponding to these bases has length r. The area of the triangle is sum of the areas of the three smaller triangles, thus:
Area of the triangle = [(21 x r)/2] + [(72 x r)/2] + [(75 x r)/2]
On the other hand, the triangle is a right triangle; thus, the area of the triangle is also equal to (21 x 72)/2. Let’s set the two expressions for the area equal to each other:
(21 x 72)/2 = [(21 x r)/2] + [(72 x r)/2] + [(75 x r)/2]
Let’s multiply each side by 2 and then solve for r:
21 x 72 = (21 x r) + (72 x r) + (75 x r)
21r + 72r + 75r = 21 x 72
168r = 21 x 72
r = (21 x 72)/168 = 72/8 = 9
Answer: E _________________
★
★
★
★
★
250 REVIEWS
5-STAR RATED ONLINE GMAT QUANT SELF STUDY COURSE
NOW WITH GMAT VERBAL (BETA)
See why Target Test Prep is the top rated GMAT quant course on GMAT Club. Read Our Reviews