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Difference between revisions of "2017 AMC 10B Problems/Problem 15"

(Created page with "==Problem== Placeholder ==Solution== Placeholder ==See Also== {{AMC10 box|year=2017|ab=B|num-b=14|num-a=16}} {{MAA Notice}}")
 
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==Problem==
 
==Problem==
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Placeholder
+
Rectangle <math>ABCD</math> has <math>AB=3</math> and <math>BC=4</math>. Point <math>E</math> is the foot of the perpendicular from <math>B</math> to diagonal <math>\overline{AC}</math>. What is the area of <math>\triangle AED</math>?
 +
 
 +
<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ \frac{42}{25}\qquad\textbf{(C)}\ \frac{28}{15}\qquad\textbf{(D)}\ 2\qquad\textbf{(E)}\ \frac{54}{25}</math>
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 +
==Solution 1==
 +
 
 +
<asy>
 +
pair A,B,C,D,E;
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A=(0,4);
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B=(3,4);
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C=(3,0);
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D=(0,0);
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draw(A--B--C--D--cycle);
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label("$A$",A,N);
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label("$B$",B,N);
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label("$C$",C,S);
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label("$D$",D,S);
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E=foot(B,A,C);
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draw(E--B);
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draw(A--C);
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draw(rightanglemark(B,E,C));
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label("$E$",E,N);
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draw(D--E);
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label("$3$",A--B,N);
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label("$4$",B--C,E);
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</asy>
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 +
First, note that <math>AC=5</math> because <math>ABC</math> is a right triangle. In addition, we have <math>AB\cdot BC=2[ABC]=AC\cdot BE</math>, so <math>BE=\frac{12}{5}</math>. Using similar triangles within <math>ABC</math>, we get that <math>AE=\frac{9}{5}</math> and <math>CE=\frac{16}{5}</math>.
 +
 
 +
Let <math>F</math> be the foot of the perpendicular from <math>E</math> to <math>AB</math>. Since <math>EF</math> and <math>BC</math> are parallel, <math>\Delta AFE</math> is similar to <math>\Delta ABC</math>. Therefore, we have <math>\frac{AF}{AB}=\frac{AE}{AC}=\frac{9}{25}</math>. Since <math>AB=3</math>, <math>AF=\frac{27}{25}</math>. Note that <math>AF</math> is an altitude of <math>\Delta AED</math> from <math>AD</math>, which has length <math>4</math>. Therefore, the area of <math>\Delta AED</math> is <math>\frac{1}{2}\cdot\frac{27}{25}\cdot4=\boxed{\textbf{(E)}\frac{54}{25}}.</math>
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==Solution 2==
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From similar triangles, we know that <math>AE=\frac{9}{5}</math> (see Solution 1). Furthermore, we also know that <math>AD=4</math> from the rectangle. Using the sine formula for area, we have
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<cmath>\frac{1}{2}(AE)(AD)(\sin(m\angle EAD)).</cmath>
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But, note that <math>\sin(m\angle EAD)=\sin(m\angle CAD)=\frac{O}{H}=\frac{3}{5}</math>. Thus, we see that
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<cmath>[AED]=\frac{1}{2}\cdot \frac{9}{5}\cdot 4\cdot\frac{3}{5}=\boxed{\textbf{(E)}\frac{54}{25}}.</cmath>
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~coolwiz
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==Solution 3==
 +
Alternatively, we can use coordinates. Denote <math>D</math> as the origin. We find the equation for <math>AC</math> as <math>y=-\frac{4}{3}x+4</math>, and <math>BE</math> as <math>y=\frac{3}{4}x+\frac{7}{4}</math>. Solving for <math>x</math> yields <math>\frac{27}{25}</math>. Our final answer then becomes <math>\frac{1}{2}\cdot\frac{27}{25}\cdot4=\boxed{\textbf{(E)}\frac{54}{25}}.</math>
 +
 
 +
==Solution 4==
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We note that the area of <math>ABE</math> must equal the area of <math>AED</math> because they share the base  <math>AE</math> and the height of both is the altitude of congruent triangles. Therefore, we find the area of <math>ABE</math> to be <math>\frac{1}{2}*\frac{9}{5}*\frac{12}{5}=\boxed{\textbf{(E)}\frac{54}{25}}.</math>
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==Solution 5==
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We know all right triangles are 5-4-3, so the areas are proportional to the square of corresponding sides. Area of <math>ABE</math> is <math> (\dfrac{3}{5})^2</math> of <math>ABC = \frac{54}{25}</math>. Using similar logic in Solution 4, Area of <math>AED</math> is the same as <math>ABE</math>.
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==Solution 6==
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Drop an altitude from <math>E</math> to <math>BC</math> and call its foot <math>X</math>. We have that <math>EX \cdot BC=BE \cdot EC</math> since both are equal to two times the area of <math>BEC</math>. Since <math>BC=4</math>, <math>BE=\frac{12}{5}</math>, and <math>EC=\frac{16}{5}</math>, we can calculate that <math>EX=\frac{48}{25}</math>. If <math>EX</math> is extended to meet <math>AD</math> at point <math>Y</math>, <math>EY=3-\frac{48}{25}=\frac{27}{25}</math>. Therefore, <math>[AED]=\frac{EY \cdot AD}{2}=\frac{\frac{27}{25} \cdot 4}{2}=\boxed{\textbf{(E)}\frac{54}{25}}</math>.
 +
 
 +
- Fasolinka
 +
 
 +
== Video Solution by OmegaLearn ==
 +
https://youtu.be/GrCtzL0S-Uo?t=369
 +
 
 +
~ pi_is_3.14
 +
 
 +
==Video Solution by Math4All999==
 +
https://www.youtube.com/watch?v=hSrgGKxduho
 +
 
 +
~Math4All999
  
−
==Solution==
 
−
Placeholder
 
 
==See Also==
 
==See Also==
 
{{AMC10 box|year=2017|ab=B|num-b=14|num-a=16}}
 
{{AMC10 box|year=2017|ab=B|num-b=14|num-a=16}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 00:20, 4 December 2023

Problem

Rectangle $ABCD$ has $AB=3$ and $BC=4$. Point $E$ is the foot of the perpendicular from $B$ to diagonal $\overline{AC}$. What is the area of $\triangle AED$?

$\textbf{(A)}\ 1\qquad\textbf{(B)}\ \frac{42}{25}\qquad\textbf{(C)}\ \frac{28}{15}\qquad\textbf{(D)}\ 2\qquad\textbf{(E)}\ \frac{54}{25}$

Solution 1

[asy] pair A,B,C,D,E; A=(0,4); B=(3,4); C=(3,0); D=(0,0); draw(A--B--C--D--cycle); label("$A$",A,N); label("$B$",B,N); label("$C$",C,S); label("$D$",D,S); E=foot(B,A,C); draw(E--B); draw(A--C); draw(rightanglemark(B,E,C)); label("$E$",E,N); draw(D--E); label("$3$",A--B,N); label("$4$",B--C,E); [/asy]

First, note that $AC=5$ because $ABC$ is a right triangle. In addition, we have $AB\cdot BC=2[ABC]=AC\cdot BE$, so $BE=\frac{12}{5}$. Using similar triangles within $ABC$, we get that $AE=\frac{9}{5}$ and $CE=\frac{16}{5}$.

Let $F$ be the foot of the perpendicular from $E$ to $AB$. Since $EF$ and $BC$ are parallel, $\Delta AFE$ is similar to $\Delta ABC$. Therefore, we have $\frac{AF}{AB}=\frac{AE}{AC}=\frac{9}{25}$. Since $AB=3$, $AF=\frac{27}{25}$. Note that $AF$ is an altitude of $\Delta AED$ from $AD$, which has length $4$. Therefore, the area of $\Delta AED$ is $\frac{1}{2}\cdot\frac{27}{25}\cdot4=\boxed{\textbf{(E)}\frac{54}{25}}.$

Solution 2

From similar triangles, we know that $AE=\frac{9}{5}$ (see Solution 1). Furthermore, we also know that $AD=4$ from the rectangle. Using the sine formula for area, we have \[\frac{1}{2}(AE)(AD)(\sin(m\angle EAD)).\] But, note that $\sin(m\angle EAD)=\sin(m\angle CAD)=\frac{O}{H}=\frac{3}{5}$. Thus, we see that \[[AED]=\frac{1}{2}\cdot \frac{9}{5}\cdot 4\cdot\frac{3}{5}=\boxed{\textbf{(E)}\frac{54}{25}}.\] ~coolwiz

Solution 3

Alternatively, we can use coordinates. Denote $D$ as the origin. We find the equation for $AC$ as $y=-\frac{4}{3}x+4$, and $BE$ as $y=\frac{3}{4}x+\frac{7}{4}$. Solving for $x$ yields $\frac{27}{25}$. Our final answer then becomes $\frac{1}{2}\cdot\frac{27}{25}\cdot4=\boxed{\textbf{(E)}\frac{54}{25}}.$

Solution 4

We note that the area of $ABE$ must equal the area of $AED$ because they share the base $AE$ and the height of both is the altitude of congruent triangles. Therefore, we find the area of $ABE$ to be $\frac{1}{2}*\frac{9}{5}*\frac{12}{5}=\boxed{\textbf{(E)}\frac{54}{25}}.$

Solution 5

We know all right triangles are 5-4-3, so the areas are proportional to the square of corresponding sides. Area of $ABE$ is $(\dfrac{3}{5})^2$ of $ABC = \frac{54}{25}$. Using similar logic in Solution 4, Area of $AED$ is the same as $ABE$.

Solution 6

Drop an altitude from $E$ to $BC$ and call its foot $X$. We have that $EX \cdot BC=BE \cdot EC$ since both are equal to two times the area of $BEC$. Since $BC=4$, $BE=\frac{12}{5}$, and $EC=\frac{16}{5}$, we can calculate that $EX=\frac{48}{25}$. If $EX$ is extended to meet $AD$ at point $Y$, $EY=3-\frac{48}{25}=\frac{27}{25}$. Therefore, $[AED]=\frac{EY \cdot AD}{2}=\frac{\frac{27}{25} \cdot 4}{2}=\boxed{\textbf{(E)}\frac{54}{25}}$.

- Fasolinka

Video Solution by OmegaLearn

https://youtu.be/GrCtzL0S-Uo?t=369

~ pi_is_3.14

Video Solution by Math4All999

https://www.youtube.com/watch?v=hSrgGKxduho

~Math4All999

See Also

2017 AMC 10B (Problems • Answer Key • Resources)
Preceded by
Problem 14 		Followed by
Problem 16
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

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