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

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

@@ Line 42: / Line 42: @@
 ==Solution 4==
-We note that the area of <math>ABE</math> must equal the area of <math>AED</math> because they share the base 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>
+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>
 ==Solution 5==
-We know all right triangles are 5-4-3, so the areas are proportional to the square of like 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>.
+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>.
+==Solution 6==
+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
 ==See Also==
 {{AMC10 box|year=2017|ab=B|num-b=14|num-a=16}}
 {{MAA Notice}}

2017 AMC 10B (Problems • Answer Key • Resources)
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