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2022 AMC 10B Problems/Problem 2: Difference between revisions

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In rhombus <math>ABCD</math>, point <math>P</math> lies on segment <math>\overline{AD}</math> so that <math>\overline{BP}</math> <math>\perp</math> <math>\overline{AD}</math>, <math>AP = 3</math>, and <math>PD = 2</math>. What is the area of <math>ABCD</math>? (Note: The figure is not drawn to scale.)
In rhombus <math>ABCD</math>, point <math>P</math> lies on segment <math>\overline{AD}</math> so that <math>\overline{BP}</math> <math>\perp</math> <math>\overline{AD}</math>, <math>AP = 3</math>, and <math>PD = 2</math>. What is the area of <math>ABCD</math>? (Note: The figure is not drawn to scale.)
(Figure redrawn to scale)
<asy>
pair A = (0,0);
label("$A$", A, SW);
pair B = (2.25,3);
label("$B$", B, NW);
pair C = (6,3);
label("$C$", C, NE);
pair D = (3.75,0);
label("$D$", D, SE);
pair P = (2.25,0);
label("$P$", P, S);
draw(A--B--C--D--cycle);
draw(P--B);
draw(rightanglemark(B,P,D));
</asy>


<math>\textbf{(A) }3\sqrt{5}\qquad\textbf{(B) }10\qquad\textbf{(C) }6\sqrt{5}\qquad\textbf{(D) }20\qquad\textbf{(E) }25</math>
<math>\textbf{(A) }3\sqrt{5}\qquad\textbf{(B) }10\qquad\textbf{(C) }6\sqrt{5}\qquad\textbf{(D) }20\qquad\textbf{(E) }25</math>

Revision as of 15:52, 17 November 2022

Problem

In rhombus $ABCD$, point $P$ lies on segment $\overline{AD}$ so that $\overline{BP}$ $\perp$ $\overline{AD}$, $AP = 3$, and $PD = 2$. What is the area of $ABCD$? (Note: The figure is not drawn to scale.)

(Figure redrawn to scale) [asy] pair A = (0,0); label("$A$", A, SW); pair B = (2.25,3); label("$B$", B, NW); pair C = (6,3); label("$C$", C, NE); pair D = (3.75,0); label("$D$", D, SE); pair P = (2.25,0); label("$P$", P, S); draw(A--B--C--D--cycle); draw(P--B); draw(rightanglemark(B,P,D)); [/asy]

$\textbf{(A) }3\sqrt{5}\qquad\textbf{(B) }10\qquad\textbf{(C) }6\sqrt{5}\qquad\textbf{(D) }20\qquad\textbf{(E) }25$

Solution

$AD = AP + PD = 3 + 2 =5$

$ABCD$ is a rhombus, so $AD = AB = 5$

$\bigtriangleup APB$ is a 3-4-5 right triangle, so $BP = 4$.

Area of a rhombus $= bh = (AD)(BP) = 5 * 4 = \boxed{\textbf{(D) }20}$.


-richiedelgado

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