What is the radius of the inscribed circle to a triangle whose sides

Question

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

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anjanita · Answer

The answer is E. Please find below my two cents

We know that, 75,72 and 21 are Pythagorean triplets
i.e., 75^2 = 72^2 +21^2
 
Length of Hypotenuse =75
Lengths of other sides is 72 and 21

We have the formula for a right angled triangle,

inradius = (Sum of Perpendicular Sides− Hypotenuse)/2 
 
inradius = ((72+21)− 75)/2

inradius = (93− 75)/2 
 
inradius = 18/2 = 9​

GMATinsight · Answer

21^2 + 72^2 = 75^2

i.e Given sides form a right triangle.

Radius of an inscribed circle,  r = \frac{∆}{S}

where ∆ = Area of Triangle  = (\frac{1}{2})*21*72 = 756 
and s = semiperimeter  = \frac{(a+b+c)}{2} = \frac{(21+72+75)}{2} = 84

i.e.  Radius of an inscribed circle,  r = \frac{756}{84} = 9

ANswer: Option E

ScottTargetTestPrep · Answer

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 =   +   +

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 =   +   +

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

Fdambro294 · Answer

Given that the triangle is a Multiple of the 7 - 24 - 25 Pythagorean triplet, we have a right triangle.

The 2 perpendicular bases are 21 and 72 and the hypotenuse is 75.

Draw the inscribed circle.

Connect the center of the inscribed circle to each of the Bases of the right triangle. The radius will be perpendicular to each side. This will create a small square with side length = r

Rule: from a single external point, if you draw 2 Lines that are tangent to the circle, each of those lines will have equal length to their respective points of tangency.

If r is the length of a portion of each Base Side, then the remainder length of each Base is:

21 - r

And

72 - r

If you look at the triangle and the inscribed circle, you will see that from the 2 non 90 degree vertices there are two lines drawn tangent to a circle from a single external point.

Thus the hypotenuse will equal the sum of the 2 above lengths:

75 = (21 - r) + (72 - r)

2r = 93 - 75

2r = 18

r = Inradius of the inscribed circle = 9

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What is the radius of the inscribed circle to a triangle whose sides

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What is the radius of the inscribed circle to a triangle whose sides

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