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

Latest revision as of 16:14, 29 June 2023

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 1

$[asy] pair A = (0,0); label("$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$", P, S); draw(A--B--C--D--cycle); draw(P--B); draw(rightanglemark(B,P,D)); [/asy]$

$\textbf{Figure redrawn to scale.}$

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

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

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

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

~richiedelgado

Solution 2 (The Area Of A Triangle)

$[asy] pair A = (0,0); label("$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$", P, S); draw(A--B--C--D--cycle); draw(D--B); draw(B--P); draw(rightanglemark(B,P,D)); [/asy]$

The diagram is from as Solution 1, but a line is constructed at $BD$ .

When it comes to the sides of a rhombus, their opposite sides are congruent and parallel. This means that $\angle ABD \cong \angle BDC$ , by the Alternate Interior Angles Theorem.

By SAS Congruence, we get $\triangle ABD \cong \triangle BDC$ .

Since $AP=3$ and $AB=5$ , we know that $BP=4$ because $\triangle APB$ is a 3-4-5 right triangle, as stated in Solution 1.

The area of $\triangle ABD$ would be $10$ , since the area of the triangle is $\frac{bh}{2}$ .

Since we know that $\triangle ABD \cong \triangle BDC$ and that $ABCD=\triangle ABD + \triangle BDC$ , so we can double the area of $\triangle ADB$ to get $10 \cdot 2 = \boxed{\textbf{(D) }20}$ .

~ghfhgvghj10, minor edits by MinecraftPlayer404

Video Solution 1

https://youtu.be/Io_GhJ6Zr_U

~Education, the Study of Everything

Video Solution(1-16)

https://youtu.be/SCwQ9jUfr0g

~~Hayabusa1

Video Solution by Interstigation

https://youtu.be/_KNR0JV5rdI?t=97

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.

@@ Line 19: / Line 19: @@
 </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>
-==Solution==
+==Solution 1==
 <asy>
@@ Line 39: / Line 43: @@
 </asy>
-(Figure redrawn to scale.)
+<cmath>\textbf{Figure redrawn to scale.}</cmath>
 <math>AD = AP + PD = 3 + 2 = 5</math>.
-<math>ABCD</math> is a rhombus, so <math>AB = AP = 5</math>.
+<math>ABCD</math> is a rhombus, so <math>AB = AD = 5</math>.
-<math>\bigtriangleup APB</math> is a 3-4-5 right triangle, so <math>BP = 4</math>.
+<math>\bigtriangleup APB</math> is a <math>3-4-5</math> right triangle, hence <math>BP = 4</math>.
-The area of the rhombus <math>= bh = (AD)(BP) = 5 \cdot 4 = \boxed{\textbf{(D) }20}</math>.
+The area of the rhombus is base times height: <math>bh = (AD)(BP) = 5 \cdot 4 = \boxed{\textbf{(D) }20}</math>.
 ~richiedelgado
+==Solution 2 (The Area Of A Triangle)==
+<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(D--B);
+draw(B--P);
+draw(rightanglemark(B,P,D));
+</asy>
+The diagram is from as Solution 1,  but a line is constructed at <math>BD</math>.
+When it comes to the sides of a rhombus, their opposite sides are congruent and parallel. This means that <math>\angle ABD \cong \angle BDC</math>, by the Alternate Interior Angles Theorem.
+By SAS Congruence, we get <math>\triangle ABD \cong \triangle BDC</math>.
+Since <math>AP=3</math> and <math>AB=5</math>, we know that <math>BP=4</math> because <math>\triangle APB</math> is a 3-4-5 right triangle, as stated in Solution 1.
+The area of <math>\triangle ABD</math> would be <math>10</math>, since the area of the triangle is <math>\frac{bh}{2}</math>.
+Since we know that <math>\triangle ABD \cong \triangle BDC</math> and that <math>ABCD=\triangle ABD + \triangle BDC</math>, so we can double the area of <math>\triangle ADB</math> to get <math>10 \cdot 2 = \boxed{\textbf{(D) }20}</math>.
+~ghfhgvghj10, minor edits by MinecraftPlayer404
+==Video Solution 1==
+https://youtu.be/Io_GhJ6Zr_U
+~Education, the Study of Everything
+==Video Solution(1-16)==
+https://youtu.be/SCwQ9jUfr0g
+~~Hayabusa1
+==Video Solution by Interstigation==
+https://youtu.be/_KNR0JV5rdI?t=97
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