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Difference between revisions of "2022 AMC 10B Problems/Problem 2"

(→‎Solution 2)
(→‎Solution)
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~richiedelgado
 
~richiedelgado
 +
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==Solution 2 ==
 +
<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));
 +
draw)
 +
</asy>
  
 
== See Also ==
 
== See Also ==

Revision as of 00:57, 18 November 2022

The following problem is from both the 2022 AMC 10B #2 and 2022 AMC 12B #2, so both problems redirect to this page.

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.)

[asy] import olympiad; size(180); real r = 3, s = 5, t = sqrt(r*r+s*s); defaultpen(linewidth(0.6) + fontsize(10)); pair A = (0,0), B = (r,s), C = (r+t,s), D = (t,0), P = (r,0); draw(A--B--C--D--A^^B--P^^rightanglemark(B,P,D)); label("$A$",A,SW); label("$B$", B, NW); label("$C$",C,NE); label("$D$",D,SE); label("$P$",P,S); [/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

[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]

(Figure redrawn to scale.)

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

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

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

The area of the rhombus $= bh = (AD)(BP) = 5 \cdot 4 = \boxed{\textbf{(D) }20}$.

~richiedelgado

Solution 2

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));
draw)
 (Error making remote request. Unknown error_msg)

See Also

2022 AMC 10B (Problems • Answer Key • Resources)
Preceded by
Problem 1 		Followed by
Problem 3
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions
2022 AMC 12B (Problems • Answer Key • Resources)
Preceded by
Problem 1 		Followed by
Problem 3
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

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