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

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Senior Manager
Joined: 30 Aug 2003
Posts: 322
Location: dallas , tx
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

Director
Joined: 28 Oct 2003
Posts: 501
Location: 55405

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27 Jan 2004, 13:25
That's an unusual Pythagorean triple, I say...
Senior Manager
Joined: 30 Aug 2003
Posts: 322
Location: dallas , tx

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27 Jan 2004, 13:36
i know.. ..but what abt radius??
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shubhangi

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Joined: 26 Dec 2003
Posts: 227
Location: India

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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.
Manager
Joined: 11 Oct 2003
Posts: 102
Location: USA

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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.
Manager
Joined: 25 Jan 2004
Posts: 92
Location: China

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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
Manager
Joined: 11 Oct 2003
Posts: 102
Location: USA

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27 Jan 2004, 21:14
I wonder how many others got it thru Zhung's way.

Director
Joined: 13 Nov 2003
Posts: 789
Location: BULGARIA

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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
Manager
Joined: 11 Oct 2003
Posts: 102
Location: USA

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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.
Intern
Joined: 20 Jan 2004
Posts: 19
Location: KZ

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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.
Manager
Joined: 11 Oct 2003
Posts: 102
Location: USA

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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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