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Difference between revisions of "2017 AMC 12B Problems/Problem 18"
(→‎Solution 5 (No sqrts))
 
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<math>\textbf{(A)}\ \frac{120}{37}\qquad\textbf{(B)}\ \frac{140}{39}\qquad\textbf{(C)}\ \frac{145}{39}\qquad\textbf{(D)}\ \frac{140}{37}\qquad\textbf{(E)}\ \frac{120}{31}</math>
 
<math>\textbf{(A)}\ \frac{120}{37}\qquad\textbf{(B)}\ \frac{140}{39}\qquad\textbf{(C)}\ \frac{145}{39}\qquad\textbf{(D)}\ \frac{140}{37}\qquad\textbf{(E)}\ \frac{120}{31}</math>
  
−
==Solution==
 
  
−
[asy]
+
==Solution 1==
 +
 
 +
<asy>
 
  /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */
 
  /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */
 
import graph; size(8.865514650638614cm);  
 
import graph; size(8.865514650638614cm);  
Line 28: Line 29:
 
  /* dots and labels */
 
  /* dots and labels */
 
dot((0.,0.),dotstyle);  
 
dot((0.,0.),dotstyle);  
−
label("<math>O</math>", (-0.10330578512396349,-0.39365890308038826), NE * labelscalefactor);  
+
label("$O$", (-0.10330578512396349,-0.39365890308038826), NE * labelscalefactor);  
 
dot((-2.,0.),dotstyle);  
 
dot((-2.,0.),dotstyle);  
−
label("<math>A</math>", (-2.2370398196844437,-0.42371149511645134), NE * labelscalefactor);  
+
label("$A$", (-2.2370398196844437,-0.42371149511645134), NE * labelscalefactor);  
 
dot((2.,0.),dotstyle);  
 
dot((2.,0.),dotstyle);  
−
label("<math>B</math>", (2.045454545454548,-0.36360631104432517), NE * labelscalefactor);  
+
label("$B$", (2.045454545454548,-0.36360631104432517), NE * labelscalefactor);  
 
dot((5.,0.),dotstyle);  
 
dot((5.,0.),dotstyle);  
−
label("<math>D</math>", (4.900450788880542,-0.42371149511645134), NE * labelscalefactor);  
+
label("$D$", (4.900450788880542,-0.42371149511645134), NE * labelscalefactor);  
 
dot((5.,5.),dotstyle);  
 
dot((5.,5.),dotstyle);  
−
label("<math>E</math>", (5.06574004507889,5.15104432757325), NE * labelscalefactor);  
+
label("$E$", (5.06574004507889,5.15104432757325), NE * labelscalefactor);  
 
dot((0.6486486486486486,1.8918918918918919),linewidth(3.pt) + dotstyle);  
 
dot((0.6486486486486486,1.8918918918918919),linewidth(3.pt) + dotstyle);  
−
label("<math>C</math>", (0.48271975957926694,2.100706235912847), NE * labelscalefactor);  
+
label("$C$", (0.48271975957926694,2.100706235912847), NE * labelscalefactor);  
 
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);  
 
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);  
 
  /* end of picture */
 
  /* end of picture */
−
[/asy]
+
</asy>
  
 
Let <math>O</math> be the center of the circle. Note that <math>EC + CA = EA = \sqrt{AD^2 + DE^2} = \sqrt{(2+2+3)^2 + 5^2} = \sqrt{74}</math>. However, by Power of a Point, <math>(EC)(EC + CA) = EO^2 - R^2 = (2+3)^2 + 5^2 - 2^2 = 25 + 25 - 4 = 46 \implies EC = \frac{46}{\sqrt{74}}</math>, so <math>AC = \sqrt{74} - \frac{46}{\sqrt{74}} = \frac{28}{\sqrt{74}}</math>. Now <math>BC = \sqrt{AB^2 - AC^2} = \sqrt{4^2 - \frac{28^2}{74}} = \sqrt{\frac{16 \cdot 74 - 28^2}{74}} = \sqrt{\frac{1184 - 784}{74}} = \frac{20}{\sqrt{74}}</math>. Since <math>\angle ACB = 90^{\circ}, [ABC] = \frac{1}{2} \cdot BC \cdot AC = \frac{1}{2} \cdot \frac{20}{\sqrt{74}} \cdot \frac{28}{\sqrt{74}} = \boxed{\textbf{(D)}\ \frac{140}{37}}</math>.
 
Let <math>O</math> be the center of the circle. Note that <math>EC + CA = EA = \sqrt{AD^2 + DE^2} = \sqrt{(2+2+3)^2 + 5^2} = \sqrt{74}</math>. However, by Power of a Point, <math>(EC)(EC + CA) = EO^2 - R^2 = (2+3)^2 + 5^2 - 2^2 = 25 + 25 - 4 = 46 \implies EC = \frac{46}{\sqrt{74}}</math>, so <math>AC = \sqrt{74} - \frac{46}{\sqrt{74}} = \frac{28}{\sqrt{74}}</math>. Now <math>BC = \sqrt{AB^2 - AC^2} = \sqrt{4^2 - \frac{28^2}{74}} = \sqrt{\frac{16 \cdot 74 - 28^2}{74}} = \sqrt{\frac{1184 - 784}{74}} = \frac{20}{\sqrt{74}}</math>. Since <math>\angle ACB = 90^{\circ}, [ABC] = \frac{1}{2} \cdot BC \cdot AC = \frac{1}{2} \cdot \frac{20}{\sqrt{74}} \cdot \frac{28}{\sqrt{74}} = \boxed{\textbf{(D)}\ \frac{140}{37}}</math>.
 +
 +
 +
==Solution 2: Similar triangles with Pythagorean==
 +
<math>AB</math> is the diameter of the circle, so <math>\angle ACB</math> is a right angle, and therefore by AA similarity, <math>\triangle ACB \sim \triangle ADE</math>.
 +
 +
Because of this, <math>\frac{AC}{AD} = \frac{AB}{AE} \Longrightarrow \frac{AC}{2+2+3} = \frac{2+2}{\sqrt{7^2 + 5^2}}</math>, so <math>AC = \frac{28}{\sqrt{74}}</math>.
 +
 +
Likewise, <math>\frac{BC}{ED} = \frac{AB}{AE} \Longrightarrow \frac{BC}{5} = \frac{4}{\sqrt{74}}</math>, so <math>BC = \frac{20}{\sqrt{74}}</math>.
 +
 +
Thus the area of <math>\triangle ABC = \frac{1}{2} \cdot \frac{28}{\sqrt{74}} \cdot \frac{20}{\sqrt{74}} = \boxed{\textbf{(D)}\ \frac{140}{37}}</math>.
 +
 +
== Solution 2b: Area shortcut ==
 +
 +
Because <math>AE</math> is <math>\sqrt{74}</math> and <math>AB</math> is <math>4</math>, the ratio of the sides is <math>\frac{\sqrt{74}}{4}</math>, meaning the ratio of the areas is thus <math>{(\frac{\sqrt{74}}{4})}^2 \implies \frac{74}{16} \implies \frac{37}{8}</math>. We then have the proportion <math>\frac{\frac{5*7}{2}}{[ABC]}=\frac{37}{8} \implies 37*[ABC]=140 \implies \boxed{\textbf{(D)}\ \frac{140}{37}}</math>
 +
 +
==Solution 3: Similar triangles without Pythagorean==
 +
Or, use similar triangles all the way, dispense with Pythagorean, and go for minimal calculation:
 +
 +
Draw <math>BF \parallel ED</math> with <math>F</math> on <math>AE</math>. <math>BF=5\times\frac{4}{7}=\frac{20}{7}</math>.
 +
 +
<math>[\triangle ABF]=\frac{1}{2} \times 4 \times \frac{20}{7}=\frac{40}{7}</math>.
 +
 +
<math>AC:CB:CF=49:35:25</math>.  (<math>7:5</math> ratio applied twice)
 +
 +
<math>[\triangle ABC]=\frac{49}{49+25}[\triangle ABF]=\boxed{\textbf{(D)}\ \frac{140}{37}}</math>.
 +
 +
==Solution 4 (Coordinate Geometry)==
 +
Let <math>A</math> be at the origin <math>(0, 0)</math> of a coordinate plane, with <math>B</math> being located at <math>(4, 0)</math>, etc.
 +
 +
We can find the area of <math>\triangle ABC</math> by finding the the altitude from line <math>AB</math> to point <math>C</math>. Realize that this altitude is the <math>y</math> coordinate of point <math>C</math> on the coordinate plane, since the respective base of <math>\triangle ABC</math> is on the <math>x</math>-axis.
 +
 +
Using the diagram in solution one, the equation for circle <math>O</math> is <math>(x-2)^2+y^2 = 4</math>.
 +
 +
The equation for line <math>AE</math> is then <math>y = \frac{5}{7}x</math>, therefore <math>x = \frac{7}{5}y</math>.
 +
 +
Substituting <math>\frac{7}{5}y</math> for <math>x</math> in the equation for circle <math>O</math>, we get:
 +
 +
<math>\left(\frac{7}{5}y-2\right)^2+y^2 = 4</math>
 +
 +
We can solve for <math>y</math> to yield the <math>y</math> coordinate of point <math>C</math> in the coordinate plane, since this is the point of intersection of the circle and line <math>AE</math>. Note that one root will yield the intersection of the circle and line <math>AE</math> at the origin, so we will ignore this root.
 +
 +
Expanding the expression and factoring, we get:
 +
 +
<math>\left(\frac{49}{25}y^2-\frac{28}{5}y+4\right)+y^2 = 4</math>
 +
 +
<math>\frac{74}{25}y^2-\frac{28}{5}y = 0</math>
 +
 +
<math>50y(37y-70) = 0</math>
 +
 +
Our non-zero root is thus <math>\frac{70}{37}</math>. Calculating the area of <math>\triangle ABC</math> with <math>4</math> as the length of <math>AB</math> and <math>\frac{70}{37}</math> as the altitude, we get:
 +
 +
<math>\frac{(4)(\frac{70}{37})}{2} = \boxed{\textbf{(D)}\ \frac{140}{37}}</math>.
 +
 +
-Solution by Joeya
 +
 +
==Solution 5 (No sqrts)==
 +
Slope of AC is 5/7
 +
As stated in other solutions AB is the diameter, ABC is right.
 +
 +
Let CF be an altitude of ABC.
 +
 +
AF:CF = CF:BF = 7:5
 +
 +
We can set AF = 49, CF = 35, BF = 25 and scale back later
 +
 +
Then the radius is <math>\frac{AB}{2}</math> = <math>\frac{AF+BF}{2}</math> = <math>\frac{74}{2}</math> = <math>37</math>.
 +
 +
So the radius is 37 and the height of ABC is 35.
 +
 +
If we scale it back so that our radius is 2, our height is <math>\frac{70}{37}</math>.
 +
 +
Area of ABC is bh/2 = <math>\frac{(4)(\frac{70}{37})}{2}</math> = <math>\boxed{\textbf{(D)}\ \frac{140}{37}}</math>.
 +
 +
-mathophobia
 +
 +
== Video Solution by OmegaLearn (Similar Triangles) ==
 +
https://youtu.be/NsQbhYfGh1Q?t=512
 +
 +
~ pi_is_3.14
  
 
==See Also==
 
==See Also==
 
{{AMC12 box|year=2017|ab=B|num-b=17|num-a=19}}
 
{{AMC12 box|year=2017|ab=B|num-b=17|num-a=19}}
 
{{MAA Notice}}
 
{{MAA Notice}}
 +
 +
[[Category:Intermediate Geometry Problems]]

Latest revision as of 20:44, 30 September 2024

Problem

The diameter $AB$ of a circle of radius $2$ is extended to a point $D$ outside the circle so that $BD=3$. Point $E$ is chosen so that $ED=5$ and line $ED$ is perpendicular to line $AD$. Segment $AE$ intersects the circle at a point $C$ between $A$ and $E$. What is the area of $\triangle ABC$?

$\textbf{(A)}\ \frac{120}{37}\qquad\textbf{(B)}\ \frac{140}{39}\qquad\textbf{(C)}\ \frac{145}{39}\qquad\textbf{(D)}\ \frac{140}{37}\qquad\textbf{(E)}\ \frac{120}{31}$


Solution 1

[asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */ import graph; size(8.865514650638614cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -6.36927122464312, xmax = 11.361758076634109, ymin = -3.789601803155515, ymax = 7.420015026296013; /* image dimensions */ draw((-2.,0.)--(0.6486486486486486,1.8918918918918919)--(2.,0.)--cycle); /* draw figures */ draw(circle((0.,0.), 2.)); draw((-2.,0.)--(5.,5.)); draw((5.,5.)--(5.,0.)); draw((5.,0.)--(-2.,0.)); draw((-2.,0.)--(0.6486486486486486,1.8918918918918919)); draw((0.6486486486486486,1.8918918918918919)--(2.,0.)); draw((2.,0.)--(-2.,0.)); draw((2.,0.)--(5.,5.)); draw((0.,0.)--(5.,5.)); /* dots and labels */ dot((0.,0.),dotstyle); label("$O$", (-0.10330578512396349,-0.39365890308038826), NE * labelscalefactor); dot((-2.,0.),dotstyle); label("$A$", (-2.2370398196844437,-0.42371149511645134), NE * labelscalefactor); dot((2.,0.),dotstyle); label("$B$", (2.045454545454548,-0.36360631104432517), NE * labelscalefactor); dot((5.,0.),dotstyle); label("$D$", (4.900450788880542,-0.42371149511645134), NE * labelscalefactor); dot((5.,5.),dotstyle); label("$E$", (5.06574004507889,5.15104432757325), NE * labelscalefactor); dot((0.6486486486486486,1.8918918918918919),linewidth(3.pt) + dotstyle); label("$C$", (0.48271975957926694,2.100706235912847), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy]

Let $O$ be the center of the circle. Note that $EC + CA = EA = \sqrt{AD^2 + DE^2} = \sqrt{(2+2+3)^2 + 5^2} = \sqrt{74}$. However, by Power of a Point, $(EC)(EC + CA) = EO^2 - R^2 = (2+3)^2 + 5^2 - 2^2 = 25 + 25 - 4 = 46 \implies EC = \frac{46}{\sqrt{74}}$, so $AC = \sqrt{74} - \frac{46}{\sqrt{74}} = \frac{28}{\sqrt{74}}$. Now $BC = \sqrt{AB^2 - AC^2} = \sqrt{4^2 - \frac{28^2}{74}} = \sqrt{\frac{16 \cdot 74 - 28^2}{74}} = \sqrt{\frac{1184 - 784}{74}} = \frac{20}{\sqrt{74}}$. Since $\angle ACB = 90^{\circ}, [ABC] = \frac{1}{2} \cdot BC \cdot AC = \frac{1}{2} \cdot \frac{20}{\sqrt{74}} \cdot \frac{28}{\sqrt{74}} = \boxed{\textbf{(D)}\ \frac{140}{37}}$.


Solution 2: Similar triangles with Pythagorean

$AB$ is the diameter of the circle, so $\angle ACB$ is a right angle, and therefore by AA similarity, $\triangle ACB \sim \triangle ADE$.

Because of this, $\frac{AC}{AD} = \frac{AB}{AE} \Longrightarrow \frac{AC}{2+2+3} = \frac{2+2}{\sqrt{7^2 + 5^2}}$, so $AC = \frac{28}{\sqrt{74}}$.

Likewise, $\frac{BC}{ED} = \frac{AB}{AE} \Longrightarrow \frac{BC}{5} = \frac{4}{\sqrt{74}}$, so $BC = \frac{20}{\sqrt{74}}$.

Thus the area of $\triangle ABC = \frac{1}{2} \cdot \frac{28}{\sqrt{74}} \cdot \frac{20}{\sqrt{74}} = \boxed{\textbf{(D)}\ \frac{140}{37}}$.

Solution 2b: Area shortcut

Because $AE$ is $\sqrt{74}$ and $AB$ is $4$, the ratio of the sides is $\frac{\sqrt{74}}{4}$, meaning the ratio of the areas is thus ${(\frac{\sqrt{74}}{4})}^2 \implies \frac{74}{16} \implies \frac{37}{8}$. We then have the proportion $\frac{\frac{5*7}{2}}{[ABC]}=\frac{37}{8} \implies 37*[ABC]=140 \implies \boxed{\textbf{(D)}\ \frac{140}{37}}$

Solution 3: Similar triangles without Pythagorean

Or, use similar triangles all the way, dispense with Pythagorean, and go for minimal calculation:

Draw $BF \parallel ED$ with $F$ on $AE$. $BF=5\times\frac{4}{7}=\frac{20}{7}$.

$[\triangle ABF]=\frac{1}{2} \times 4 \times \frac{20}{7}=\frac{40}{7}$.

$AC:CB:CF=49:35:25$. ($7:5$ ratio applied twice)

$[\triangle ABC]=\frac{49}{49+25}[\triangle ABF]=\boxed{\textbf{(D)}\ \frac{140}{37}}$.

Solution 4 (Coordinate Geometry)

Let $A$ be at the origin $(0, 0)$ of a coordinate plane, with $B$ being located at $(4, 0)$, etc.

We can find the area of $\triangle ABC$ by finding the the altitude from line $AB$ to point $C$. Realize that this altitude is the $y$ coordinate of point $C$ on the coordinate plane, since the respective base of $\triangle ABC$ is on the $x$-axis.

Using the diagram in solution one, the equation for circle $O$ is $(x-2)^2+y^2 = 4$.

The equation for line $AE$ is then $y = \frac{5}{7}x$, therefore $x = \frac{7}{5}y$.

Substituting $\frac{7}{5}y$ for $x$ in the equation for circle $O$, we get:

$\left(\frac{7}{5}y-2\right)^2+y^2 = 4$

We can solve for $y$ to yield the $y$ coordinate of point $C$ in the coordinate plane, since this is the point of intersection of the circle and line $AE$. Note that one root will yield the intersection of the circle and line $AE$ at the origin, so we will ignore this root.

Expanding the expression and factoring, we get:

$\left(\frac{49}{25}y^2-\frac{28}{5}y+4\right)+y^2 = 4$

$\frac{74}{25}y^2-\frac{28}{5}y = 0$

$50y(37y-70) = 0$

Our non-zero root is thus $\frac{70}{37}$. Calculating the area of $\triangle ABC$ with $4$ as the length of $AB$ and $\frac{70}{37}$ as the altitude, we get:

$\frac{(4)(\frac{70}{37})}{2} = \boxed{\textbf{(D)}\ \frac{140}{37}}$.

-Solution by Joeya

Solution 5 (No sqrts)

Slope of AC is 5/7 As stated in other solutions AB is the diameter, ABC is right.

Let CF be an altitude of ABC.

AF:CF = CF:BF = 7:5

We can set AF = 49, CF = 35, BF = 25 and scale back later

Then the radius is $\frac{AB}{2}$ = $\frac{AF+BF}{2}$ = $\frac{74}{2}$ = $37$.

So the radius is 37 and the height of ABC is 35.

If we scale it back so that our radius is 2, our height is $\frac{70}{37}$.

Area of ABC is bh/2 = $\frac{(4)(\frac{70}{37})}{2}$ = $\boxed{\textbf{(D)}\ \frac{140}{37}}$.

-mathophobia

Video Solution by OmegaLearn (Similar Triangles)

https://youtu.be/NsQbhYfGh1Q?t=512

~ pi_is_3.14

See Also

2017 AMC 12B (Problems • Answer Key • Resources)
Preceded by
Problem 17 		Followed by
Problem 19
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All AMC 12 Problems and Solutions

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