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# What is the radius of the inscribed circle to a triangle

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What is the radius of the inscribed circle to a triangle [#permalink]

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27 Jan 2004, 13:19
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What is the radius of the inscribed circle to a triangle whose sides measure
21cm, 72cm and 75cm respectively?
(1) 9 cm
(2) 37.5 cm
(3) 28.5 cm
(4) 14.5 cm
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shubhangi

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27 Jan 2004, 13:25
That's an unusual Pythagorean triple, I say...
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27 Jan 2004, 13:36
i know.. ..but what abt radius??
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shubhangi

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27 Jan 2004, 15:48
There is a formula for this but I dont remember. as we know the area of the triangle is 1/2 base * height and so the area is 1/2*21*76 = 756 and the area of the circle is phi r^2 which should be less than 756. So of the choices 37.5 and 28.5 is ruled out. Now left with 9 and 14. These can be the answers, in exam I would opt for 9 as the area is less and also is a multiple of 3 like the sides 21,72 and 75.
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27 Jan 2004, 18:22
Zhung wrote:

3(24+7-25)/2 = 9

Can you explain?

I eliminated the choices that are greater than 21, but could not proceed further.
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27 Jan 2004, 20:39
c = hypotenuse of a right angled traingle

The sides of the right angled traingle are tangents to the incircle.

cf: intersecting tangents to a circle are equal(use similar traingles)

c = (a-r)+(b-r)
r = (a+b-c)/2
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27 Jan 2004, 21:14
I wonder how many others got it thru Zhung's way.

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28 Jan 2004, 04:34
Since this is a right triangle the area is 21x72/2=756. The area of a triangle that has a circle inscribed is S=pr ( p-half the perimeter of triangle, r-radius of inscribed circle).P=168,(21+75+72) ,p=P/2=84 Then 756/84= r =9
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28 Jan 2004, 05:10
BG,

I knew there was a correlation between perimeter, area and inscribed traingle but could not remember. Thaks for refreshing my memory.

Zhung's approach is very fundemental in case one did not know this correlation.
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28 Jan 2004, 08:37
I say for 9 sm.

Guys,
there is quick way to approximate right choice in this particular case.

You see, the diameter of incribed circle CANNOT be more than the least of the sides of a triangle. So the most value of the radius might be is 21/2=10.5 sm. Therefore, I pick (1) w/o any calculations.

General advice. See choices before rush into calculus.
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28 Jan 2004, 09:17
Solo,

Thank you. I agree with you on this given the choices. I was looking for radius < 21 and that was my error. But, what if both 9 and 10 were given in the choices?

I'd rather know of a property that helps me in the long run.
I agree with Solo   [#permalink] 28 Jan 2004, 09:17
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