2025 AIME I Problems/Problem 6
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[hide]Problem
An isosceles trapezoid has an inscribed circle tangent to each of its four sides. The radius of the circle is , and the area of the trapezoid is
. Let the parallel sides of the trapezoid have lengths
and
, with
. Find
Diagram
Solution 1
To begin with, because of tangents from the circle to the bases, the height is The formula for the area of a trapezoid is
Plugging in our known values we have
Next, we use Pitot's Theorem which states for tangential quadrilaterals
Since we are given
is an isosceles trapezoid we have
Using Pitot's we find,
Finally we can use the Pythagorean Theorem by dropping an altitude from D,
Noting that
we find,
Solution 2 (Trigonometry)
Draw angle bisectors from the bottom left vertex to the center of the circle. Call the angle formed . Drawing a line from the center of the circle to the midway point of the bottom base of the trapezoid makes a right angle, and the other angle has to be
. Then draw a line segment from the center of the circle to the top left vertex, then you have a right triangle. The smaller angle of this triangle is
. This means
. This also means
. Note that
. The area of the trapezoid is
.
.
-alwaysgonnagiveyouup
Solution 3 (Fastest formula)
Denote the radius of the inscribed circle as , and the parallel sides as
and
.
By formula, we get
, where
.
Also, by formula,
, where
.
Therefore,
Solution 4(Double-angle Formula)
Let
Since
Set
Assume
Substitute
So the final answer is
Formula reference to here: https://en.wikipedia.org/wiki/Tangential_trapezoid
~Mitsuihisashi14
~ LaTeX by alwaysgonnagiveyouup
Solution 5
The height of the trapezoid is clearly the diameter of the circle or . We let the larger base be
, and the smaller one be
. We have by the area of a trapezoid:
. Now, by Pitot's theorem, and letting the legs of the trapezoid be
, we have that
. Now, by pythag we have that
. Now, by systems of equations, we find that
, and
. Now, we have that
-jb2015007
Video Solution(Fast and Easy)
~MC
See also
2025 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 5 |
Followed by Problem 7 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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