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Difference between revisions of "2025 AIME I Problems/Problem 6"
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==Problem==
 
==Problem==
 
An isosceles trapezoid has an inscribed circle tangent to each of its four sides. The radius of the circle is <math>3</math>, and the area of the trapezoid is <math>72</math>. Let the parallel sides of the trapezoid have lengths <math>r</math> and <math>s</math>, with <math>r \neq s</math>. Find <math>r^2+s^2</math>
 
An isosceles trapezoid has an inscribed circle tangent to each of its four sides. The radius of the circle is <math>3</math>, and the area of the trapezoid is <math>72</math>. Let the parallel sides of the trapezoid have lengths <math>r</math> and <math>s</math>, with <math>r \neq s</math>. Find <math>r^2+s^2</math>
 +
==Diagram==
 +
<asy>
 +
unitsize(0.5 cm);
 +
 +
real r = 12 + 6*sqrt(3);
 +
real s = 12 - 6*sqrt(3);
 +
real h = 6; 
 +
 +
pair A = (-r/2, 0);
 +
pair B = ( r/2, 0);
 +
pair C = ( s/2, h);
 +
pair D = (-s/2, h);
 +
 +
draw(A--B--C--D--cycle);
 +
 +
pair O = (0, h/2);
 +
draw(circle(O, 3));
 +
 +
dot(A); label("$A$", A, SW);
 +
dot(B); label("$B$", B, SE);
 +
dot(C); label("$C$", C, NE);
 +
dot(D); label("$D$", D, NW);
 +
 +
dot(O);
 +
label("$O$", (0,h/2), E);
 +
 +
label("$r$", midpoint(A--B), S);
 +
label("$s$", midpoint(C--D), N);
 +
</asy>

Revision as of 17:06, 13 February 2025

Problem

An isosceles trapezoid has an inscribed circle tangent to each of its four sides. The radius of the circle is $3$, and the area of the trapezoid is $72$. Let the parallel sides of the trapezoid have lengths $r$ and $s$, with $r \neq s$. Find $r^2+s^2$

Diagram

[asy] unitsize(0.5 cm); real r = 12 + 6*sqrt(3); real s = 12 - 6*sqrt(3); real h = 6; pair A = (-r/2, 0); pair B = ( r/2, 0); pair C = ( s/2, h); pair D = (-s/2, h); draw(A--B--C--D--cycle); pair O = (0, h/2); draw(circle(O, 3)); dot(A); label("$A$", A, SW); dot(B); label("$B$", B, SE); dot(C); label("$C$", C, NE); dot(D); label("$D$", D, NW); dot(O); label("$O$", (0,h/2), E); label("$r$", midpoint(A--B), S); label("$s$", midpoint(C--D), N); [/asy]

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