2025 AIME I Problems/Problem 6: Difference between revisions
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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 
, and the area of the trapezoid is 
. Let the parallel sides of the trapezoid have lengths 
and 
, with 
. Find 
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]](http://latex.artofproblemsolving.com/a/b/5/ab5d3517b5da1faf886c4f955d32f7ff73973023.png)
