1999 AHSME Problems/Problem 16
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Problem
What is the radius of a circle inscribed in a rhombus with diagonals of length 
and 
?
Solution
Let 
and 
be the lengths of the diagonals, 
the side, and 
the radius of the inscribed circle.
Using Pythagorean theorem we can compute 
.
We can now express the area of the rhombus in two different ways: as 
, and as 
. Solving 
for we get 
.
(The first formula computes the area as one half of the circumscribed rectangle whose sides are parallel to the diagonals. The second one comes from the fact that we can divide the rhombus into 
equal triangles, and in those the height on the side is equal to . See pictures below.)
![[asy] unitsize(1cm); pair A=(-3,0), B=(0,-1.25), C=(3,0), D=(0,1.25); pair E=(-3,-1.25), F=(-3,1.25), G=(3,1.25), H=(3,-1.25); fill ( E -- F -- G -- H -- cycle, lightgray ); draw ( E -- F -- G -- H -- cycle ); draw ( A -- B -- C -- D -- cycle ); label("$d_1$",(E+F)*0.5,W); label("$d_2$",(F+G)*0.5,N); [/asy]](http://latex.artofproblemsolving.com/2/3/0/2303c6252536624af6d17519867812d65c95e642.png)
![[asy] unitsize(1cm); pair A=(-3,0), B=(0,-1.25), C=(3,0), D=(0,1.25); pair P = intersectionpoint( A--B--C--D--cycle, circle( (0,0), 15/13 ) ); fill ( A -- B -- (0,0) -- cycle, lightgray ); draw ( A -- B -- (0,0) -- cycle ); draw ( A -- B -- C -- D -- cycle ); draw ( circle( (0,0), 15/13 ) ); draw ( (0,0) -- P ); label("$a$",(A+B)*0.5,SW); label("$r$",P*0.5,0.5*NW); [/asy]](http://latex.artofproblemsolving.com/a/3/9/a39997feae4fb3b6d4af1ebe941e93a0483f720e.png)
See also
| 1999 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 15 |
Followed by Problem 17 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 | ||
| All AHSME Problems and Solutions | ||
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