2024 AIME I Problems/Problem 5
Contents
Problem
Rectangles and are drawn such that are collinear. Also, all lie on a circle. If ,,, and , what is the length of ?
Solution 1
Suppose . Extend and until they meet at . From the Power of a Point Theorem, we have . Substituting in these values, we get . Using simple guess and check, we find that so .
~alexanderruan
Solution 2
We use simple geometry to solve this problem.
We are given that , , , and are concyclic; call the circle that they all pass through circle with center . We know that, given any chord on a circle, the perpendicular bisector to the chord passes through the center; thus, given two chords, taking the intersection of their perpendicular bisectors gives the center. We therefore consider chords and and take the midpoints of and to be and , respectively.
We could draw the circumcircle, but actually it does not matter for our solution; all that matters is that , where is the circumradius.
By the Pythagorean Theorem, . Also, . We know that , and ; ; ; and finally, . Let . We now know that and . Recall that ; thus, . We solve for :
\begin{align*} (x+92)^2+8^2&=25^2+92^2 \\ (x+92)^2&=625+(100-8)^2-8^2 \\ &=625+10000-1600+64-64 \\ &=9025 \\ x+92&=95 \\ x&=3. \\ \end{align*}
The question asks for , which is .
~Technodoggo
Solution 3
We find that
Let and . By similar triangles we have . Substituting lengths we have Solving, we find and thus ~AtharvNaphade ~coolruler ~eevee9406
Solution 4
One liner:
~Bluesoul
Explanation
Let intersect at (using the same diagram as Solution 2).
The formula calculates the distance from to (or ), , then shifts it to and the finds the distance from to , . minus that gives , and when added to , half of , gives
Solution 5
Let This means that Since quadrilateral is cyclic,
Let Then, $\Delta DXA \~ \Delta FXG,$ (Error compiling LaTeX. Unknown error_msg) with side ratio Also, since Using the similar triangles, we have and
Since we want we only need to solve for in this system of equations. Solving yields so
~PureSwag
Video Solution with Circle Properties
https://youtu.be/1LWwJeFpU9Y
~Veer Mahajan
Video Solution 1 by OmegaLearn.org
See also
2024 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 4 |
Followed by Problem 6 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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