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Math Help - 3 Circles whose radii form a Triangle

  1. #1
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    3 Circles whose radii form a Triangle

    The Problem: In the diagram, the circles with centres P, Q, and R hav radii of 3, 2, and 1 respectively. Each circle touches the other two as shown. Find the area of triangle PQR.

    I used the law of cosines to find angle P.

    c^2 = a^2 + b^2 - 2ac(cos \theta)
    3^2 = 4^3 + 5^2 -2(4)(5)(cos P)
     9 = 41 -40 (cos P)
    -32 = -40 Cos P
     -32/-40 = Cos P
    0.8 = Cos P
    0.8 (inverse cos) = P
     P = 36.87^o

    I then thought of using the sine or cosine formula to find the height (I thought of drawing a height from the centre R to where the circles containing P and Q touch). But I don't know that it's a right-angle triangle. I am stuck at this point.
    Attached Thumbnails Attached Thumbnails 3 Circles whose radii form a Triangle-triangle-formed-circles.png  
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  2. #2
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    Hello, D. Martin!

    This is a problem from the Dirty Tricks division.
    . . Warning: It's a groaner!


    In the diagram, the circles with centres P, Q,R have radii of 3, 2, and 1, resp.
    Each circle touches the other two as shown.
    Find the area of \Delta PQR.

    The answer is SO awful that I'll hide it . . .


    Drag your cursor between the asterisks . . .

    *

    The centers form a 3-4-5 right triangle!

    *


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  3. #3
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    Thanks, but is it a right angle triangle because two lines two a circle centre result in a right angle iFF they both come from adjacent circles or something like that? I know there's probably a simple rule that explains it, but I can't quite put my finger on it.

    But, anyways, thank you a lot for the help. I've always considered myself horrible at spatial questions.
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