Difference between revisions of "1997 AHSME Problems/Problem 15"

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==Problem==
 
==Problem==
  
Medians <math>BD</math> and <math>AE</math> of triangle <math>ABC</math> are perpendicular, <math>BD=8</math>, and <math>CE=12</math>.  The area of triangle <math>ABC</math> is  
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Medians <math>BD</math> and <math>CE</math> of triangle <math>ABC</math> are perpendicular, <math>BD=8</math>, and <math>CE=12</math>.  The area of triangle <math>ABC</math> is  
  
 
<asy>
 
<asy>
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<math> \textbf{(A)}\ 24\qquad\textbf{(B)}\ 32\qquad\textbf{(C)}\ 48\qquad\textbf{(D)}\ 64\qquad\textbf{(E)}\ 96 </math>
 
<math> \textbf{(A)}\ 24\qquad\textbf{(B)}\ 32\qquad\textbf{(C)}\ 48\qquad\textbf{(D)}\ 64\qquad\textbf{(E)}\ 96 </math>
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==Solution==
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<asy>
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defaultpen(linewidth(.8pt));
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dotfactor=4;
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pair A = origin;
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pair B = (1.25,1);
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pair C = (2,0);
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pair D = midpoint(A--C);
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pair E = midpoint(A--B);
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pair F = midpoint(B--C);
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pair G = intersectionpoint(E--C,B--D);
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dot(A);dot(B);dot(C);dot(D);dot(E);dot(G);dot(F);
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label("$A$",A,S);label("$B$",B,N);label("$C$",C,S);label("$D$",D,S);label("$E$",E,NW);label("$G$",G,NE);label("$F$",F,NE);
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draw(A--B--C--cycle);
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draw(B--D);
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draw(E--C);
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draw(A--F);
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draw(rightanglemark(B,G,E,3));</asy>
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One median divides a triangle into <math>2</math> equal areas, so all three medians will divide a triangle into <math>6</math> equal areas.
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The median <math>CE</math> is divided into a <math>2:1</math> ratio at centroid <math>G</math>, so <math>GE = \frac{1}{3}\cdot CE = \frac{1}{3}\cdot 12 = 4</math>
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Similarly, <math>BG = \frac{2}{3}\cdot 8 = \frac{16}{3}</math>
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The area of the right triangle <math>\triangle BEG</math> is <math>\frac{1}{2}\cdot\frac{16}{3}\cdot 4</math>
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The area of the whole figure is <math>6\cdot \frac{1}{2}\cdot\frac{16}{3}\cdot 4 = 64</math>, and the correct answer is <math>\boxed{D}</math>.
  
 
== See also ==
 
== See also ==
{{AHSME box|year=1997|num-b=10|num-a=12}}
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{{AHSME box|year=1997|num-b=14|num-a=16}}
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{{MAA Notice}}

Revision as of 01:20, 15 July 2016

Problem

Medians $BD$ and $CE$ of triangle $ABC$ are perpendicular, $BD=8$, and $CE=12$. The area of triangle $ABC$ is

[asy] defaultpen(linewidth(.8pt)); dotfactor=4; pair A = origin; pair B = (1.25,1); pair C = (2,0); pair D = midpoint(A--C); pair E = midpoint(A--B); pair G = intersectionpoint(E--C,B--D); dot(A);dot(B);dot(C);dot(D);dot(E);dot(G); label("$A$",A,S);label("$B$",B,N);label("$C$",C,S);label("$D$",D,S);label("$E$",E,NW);label("$G$",G,NE); draw(A--B--C--cycle); draw(B--D); draw(E--C); draw(rightanglemark(C,G,D,3));[/asy]

$\textbf{(A)}\ 24\qquad\textbf{(B)}\ 32\qquad\textbf{(C)}\ 48\qquad\textbf{(D)}\ 64\qquad\textbf{(E)}\ 96$

Solution

[asy] defaultpen(linewidth(.8pt)); dotfactor=4; pair A = origin; pair B = (1.25,1); pair C = (2,0); pair D = midpoint(A--C); pair E = midpoint(A--B); pair F = midpoint(B--C); pair G = intersectionpoint(E--C,B--D); dot(A);dot(B);dot(C);dot(D);dot(E);dot(G);dot(F); label("$A$",A,S);label("$B$",B,N);label("$C$",C,S);label("$D$",D,S);label("$E$",E,NW);label("$G$",G,NE);label("$F$",F,NE); draw(A--B--C--cycle); draw(B--D); draw(E--C); draw(A--F); draw(rightanglemark(B,G,E,3));[/asy]

One median divides a triangle into $2$ equal areas, so all three medians will divide a triangle into $6$ equal areas.

The median $CE$ is divided into a $2:1$ ratio at centroid $G$, so $GE = \frac{1}{3}\cdot CE = \frac{1}{3}\cdot 12 = 4$

Similarly, $BG = \frac{2}{3}\cdot 8 = \frac{16}{3}$

The area of the right triangle $\triangle BEG$ is $\frac{1}{2}\cdot\frac{16}{3}\cdot 4$

The area of the whole figure is $6\cdot \frac{1}{2}\cdot\frac{16}{3}\cdot 4 = 64$, and the correct answer is $\boxed{D}$.

See also

1997 AHSME (ProblemsAnswer KeyResources)
Preceded by
Problem 14
Followed by
Problem 16
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