Difference between revisions of "2025 AIME I Problems/Problem 6"

Revision as of 07:24, 28 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$", midpoint(C--D), N); [/asy]$

Solution 1

To begin with, because of tangents from the circle to the bases, the height is $2\cdot3=6.$ The formula for the area of a trapezoid is $\frac{h(b_1+b_2)}{2}.$ Plugging in our known values we have $\frac{6(r+s)}{2}=72.$ $r+s=24.$ Next, we use Pitot's Theorem which states for tangential quadrilaterals $AB+CD=AD+BC.$ Since we are given $ABCD$ is an isosceles trapezoid we have $AD=BC=x.$ Using Pitot's we find, $AB+CD=r+s=2x=24.$ $x=12.$ Finally we can use the Pythagorean Theorem by dropping an altitude from D, $\left(\frac{r - s}{2}\right)^2 + 6^2 = 12^2.$ $\left(\frac{r-s}{2}\right)^2=108.$ $(r-s)^2=432.$ Noting that $\frac{(r + s)^2 + (r - s)^2}{2} = r^2 + s^2$ we find, $\frac{(24^2+432)}{2}=\boxed{504}$

~mathkiddus

Solution 2 (Trigonometry)

Draw angle bisectors from the bottom left vertex to the center of the circle. Call the angle formed $\theta$ . 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 $90^{\circ} - \theta$ . 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 $180^{\circ} - (180^{\circ} - \theta) = \theta$ . This means $\frac{r}{2} = \frac{3}{\tan \theta} \implies r = \frac{6}{\tan \theta}$ . This also means $\frac{s}{2} = 3 \tan \theta \implies s = 6 \tan \theta$ . Note that $r^2 + s^2 = (r + s)^2 - 2rs.$ $rs = \frac{6}{\tan \theta} \cdot 6 \tan \theta = 36 \implies 2rs = 72$ . The area of the trapezoid is $72 = 6 \cdot \frac{r + s}{2} \implies r + s = 24$ . $(r + s)^2 - 2rs = 576 - 72 = \boxed{504}$ .

$[asy] size(15cm); draw(circle((0,0), 3)); draw((-0.5 * (12 + 6sqrt(3)), -3) -- (0.5 * (12 + 6sqrt(3)), -3) -- (0.5 * (12 - 6sqrt(3)), 3) -- (-0.5 * (12 - 6sqrt(3)), 3) -- cycle); draw((0, 0) -- (-0.5 * (12 + 6*sqrt(3)),-3) -- cycle); draw((0, 0) -- (0.5 * (12 + 6*sqrt(3)),-3) -- cycle); draw((0, 0) -- (-0.5 * (12 - 6*sqrt(3)),3) -- cycle); draw((0, 0) -- (0.5 * (12 - 6*sqrt(3)),3) -- cycle); draw((0, 0) -- (0,3) -- cycle); draw((0, 0) -- (0,-3) -- cycle); draw((-0.5, -3) -- (-0.5,-2.5) -- (0.5, -2.5) -- (0.5, -3) -- cycle); draw((-0.25, 3) -- (-0.25, 2.75) -- (0.25, 2.75) -- (0.25, 3) -- cycle); label("$\theta$", (-0.5 * (12 + 6*sqrt(3)) + 3, -3), NE); label("$\theta$", (0.5 * (12 + 6*sqrt(3)) - 3, -3), NW); label("$\theta$", (-0.5 * (12 + 6*sqrt(3)) + 3, -2), NE); label("$\theta$", (0.5 * (12 + 6*sqrt(3)) - 3, -2), NW); label("$90^{\circ} - \theta$", (0, -0.5), SW); label("$90^{\circ} - \theta$", (0, -0.5), SE); label("$90^{\circ}$", (0 - 0.1, 0), NW); label("$90^{\circ}$", (0 + 0.1, 0), NE); label("$\frac{r}{2}$", (-0.25 * (12 + 6*sqrt(3)), -3), S); label("$\frac{r}{2}$", (0.25 * (12 + 6*sqrt(3)), -3), S); label("$\theta$", (-0.1, 1.75), E); label("$\theta$", (0.1, 1.75), W); label("$\frac{s}{2}$", (-0.25 * (12 - 6*sqrt(3)), 3), N); label("$\frac{s}{2}$", (0.25 * (12 - 6*sqrt(3)), 3), N); [/asy]$

-alwaysgonnagiveyouup

Solution 3 (Fastest formula)

Denote the radius of the inscribed circle as $R$ , and the parallel sides as $r$ and $s$ . By formula, we get $R = 3 = \frac{1}{2} \cdot \sqrt{rs}$ , where $rs = 36$ . Also, by formula, $A = 72 = \frac{1}{2} \cdot \sqrt{rs} \cdot (r + s)$ , where $r + s = 24$ . Therefore, \begin{align*} &r^2 + s^2 = (r + s)^2 - 2rs \\ &= 24^2 - 2 \cdot 36 \\ &= \boxed{504} \end{align*}

Solution 4(Double-angle Formula)

Let $\angle OAB = \alpha$, $\tan(\alpha)=\frac{6}{s}$. By the double - angle formula for tangent $\tan(2\alpha)=\frac{2\tan\alpha}{1-\tan^{2}\alpha}=\frac{2\times\frac{6}{s}}{1 - (\frac{6}{s})^{2}}=\frac{\frac{12}{s}}{\frac{s^{2}-36}{s^{2}}}=\frac{12s}{s^{2}-36}$.

Since $\angle DAB = 2\alpha$, $\tan(2\alpha)=\frac{12}{s - r}=\frac{12s}{s(s - r)}=\frac{12s}{s^{2}-sr}$.

Set $\frac{12s}{s^{2}-sr}=\frac{12s}{s^{2}-36}$. Since $s\neq0$, we can cancel out $12s$ from both sides of the equation, getting $s^{2}-sr=s^{2}-36$. Subtracting $s^{2}$ from both sides, we have $-sr=-36$, so $sr = 36$.

Assume $(r + s)^{2}=576$. Using the formula $(r + s)^{2}=r^{2}+2rs + s^{2}$, then $r^{2}+s^{2}=(r + s)^{2}-2rs$.

Substitute $rs = 36$ and $(r + s)^{2}=576$ into the formula: $r^{2}+s^{2}=576-2\times36=576 - 72=504$.

So the final answer is $504$.

Formula reference to here: https://en.wikipedia.org/wiki/Tangential_trapezoid

~Mitsuihisashi14

~ LaTeX by alwaysgonnagiveyouup

Video Solution(Fast and Easy)

https://youtu.be/zK5C_FkdhlM

~MC

@@ Line 85: / Line 85: @@
 ==Solution 4(Double-angle Formula)==
-Let
+Let \(\angle OAB = \alpha\), \(\tan(\alpha)=\frac{6}{s}\). By the double - angle formula for tangent \(\tan(2\alpha)=\frac{2\tan\alpha}{1-\tan^{2}\alpha}=\frac{2\times\frac{6}{s}}{1 - (\frac{6}{s})^{2}}=\frac{\frac{12}{s}}{\frac{s^{2}-36}{s^{2}}}=\frac{12s}{s^{2}-36}\).
-∠OAB=α
-,
+Since \(\angle DAB = 2\alpha\), \(\tan(2\alpha)=\frac{12}{s - r}=\frac{12s}{s(s - r)}=\frac{12s}{s^{2}-sr}\).
-tan(α)=
-s
+Set \(\frac{12s}{s^{2}-sr}=\frac{12s}{s^{2}-36}\). Since \(s\neq0\), we can cancel out \(12s\) from both sides of the equation, getting \(s^{2}-sr=s^{2}-36\). Subtracting \(s^{2}\) from both sides, we have \(-sr=-36\), so \(sr = 36\).
+Assume \((r + s)^{2}=576\). Using the formula \((r + s)^{2}=r^{2}+2rs + s^{2}\), then \(r^{2}+s^{2}=(r + s)^{2}-2rs\).
-. By the double - angle formula for tangent
+Substitute \(rs = 36\) and \((r + s)^{2}=576\) into the formula: \(r^{2}+s^{2}=576-2\times36=576 - 72=504\).
-tan(2α)=
-−tan
+So the final answer is \(504\).
- α
-tanα
- =
-−(
-s
- )
-×
-s
- =
-s
-s
- −36
-s
- =
-s
- −36
-s
-.
-Since
-∠DAB=2α
-,
-tan(2α)=
-s−r
- =
-s(s−r)
-s
- =
-s
- −sr
-s
-.
-Set
-s
- −sr
-s
- =
-s
- −36
-s
-. Since
-s
-
-=0
-, we can cancel out
-s
- from both sides of the equation, getting
-s
- −sr=s
- −36
-. Subtracting
-s
- from both sides, we have
-−sr=−36
-, so
-sr=36
-.
-Assume
-(r+s)
- =576
-. Using the formula
-(r+s)
- =r
- +2rs+s
-, then
-r
- +s
- =(r+s)
- −2rs
-.
-Substitute
-rs=36
- and
-(r+s)
- =576
- into the formula:
-r
- +s
- =576−2×36=576−72=504
-.
-So the final answer is
-.
 Formula reference to here: https://en.wikipedia.org/wiki/Tangential_trapezoid

2025 AIME I (Problems • Answer Key • Resources)
Preceded by Problem 5	Followed by Problem 7
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