Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "1984 AHSME Problems/Problem 26"
(→‎Solution)
Line 4: Line 4:
 
<math> \mathrm{(A) \ }9 \qquad \mathrm{(B) \ }12 \qquad \mathrm{(C) \ } 15 \qquad \mathrm{(D) \ }18 \qquad \mathrm{(E) \ } \text{Not uniquely determined} </math>
 
<math> \mathrm{(A) \ }9 \qquad \mathrm{(B) \ }12 \qquad \mathrm{(C) \ } 15 \qquad \mathrm{(D) \ }18 \qquad \mathrm{(E) \ } \text{Not uniquely determined} </math>
  
−
==Solution==
+
==Solution 1==
 
[[File:1984AHSME26.png]]
 
[[File:1984AHSME26.png]]
  
Line 15: Line 15:
 
Now note that <math>CE</math> is the height of triangle <math>BDE</math> originating from vertex <math>E</math>, so we have that
 
Now note that <math>CE</math> is the height of triangle <math>BDE</math> originating from vertex <math>E</math>, so we have that
  
−
<cmath>[BED]=\frac{BD\cdot CE}{2}=\frac{ac\cos{\beta}\tan{\beta}}{4}=\frac{ac\sin{\beta}}{4}</cmath>
+
<math>[BED]=\frac{BD\cdot CE}{2}=\frac{ac\cos{\beta}\tan{\beta}}{4}=\frac{ac\sin{\beta}}{4}</math>
  
−
However, this is simply half the area of triangle <math>ABC</math>, so <math>[BED]=12</math>, which makes <math>\mathrm{(B) \ }</math> the correct answer.
+
However, this is simply half the area of triangle <math>ABC</math>, so <math>[BED]=12</math>, which makes <math>\boxed{\textbf{B}}</math> the correct answer.
 +
 
 +
==Solution 2==
 +
It is well known that <math>[CMB]=12</math>. But <math>\triangle{CBE}</math>~<math>\triangle{DBM}</math>. As a result <math>[CMB]=[BED] \implies \boxed{\textbf{B}}</math>.
  
 
==See Also==
 
==See Also==
 
{{AHSME box|year=1984|num-b=25|num-a=27}}
 
{{AHSME box|year=1984|num-b=25|num-a=27}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 19:37, 12 February 2017

Problem

In the obtuse triangle $ABC$ with $\angle C>90^\circ$, $AM=MB$, $MD\perp BC$, and $EC\perp BC$ ($D$ is on $BC$, $E$ is on $AB$, and $M$ is on $EB$). If the area of $\triangle ABC$ is $24$, then the area of $\triangle BED$ is

$\mathrm{(A) \ }9 \qquad \mathrm{(B) \ }12 \qquad \mathrm{(C) \ } 15 \qquad \mathrm{(D) \ }18 \qquad \mathrm{(E) \ } \text{Not uniquely determined}$

Solution 1

1984AHSME26.png

We let side $BC$ have length $a$, $AB$ have length $c$, and $\angle ABC$ have angle measure $\beta$. We then have that

\[[ABC]=24=\frac{AB\cdot BC\sin{\angle ABC}}{2}=\frac{ac\sin{\beta}}{2}\]

Now I shall find the lengths of $BD$ and $CE$ in terms of the defined variables. Note that $M$ is defined to be the midpoint of $AB$, so $BM=\frac{c}{2}$. We can then use trigonometric manipulation on triangle $BDM$ to get that $BD=\frac{c\cos{\beta}}{2}$. We can also use trig manipulation on $BCE$ to get that $CE=a\tan{\beta}$.

Now note that $CE$ is the height of triangle $BDE$ originating from vertex $E$, so we have that

$[BED]=\frac{BD\cdot CE}{2}=\frac{ac\cos{\beta}\tan{\beta}}{4}=\frac{ac\sin{\beta}}{4}$

However, this is simply half the area of triangle $ABC$, so $[BED]=12$, which makes $\boxed{\textbf{B}}$ the correct answer.

Solution 2

It is well known that $[CMB]=12$. But $\triangle{CBE}$~$\triangle{DBM}$. As a result $[CMB]=[BED] \implies \boxed{\textbf{B}}$.

See Also

1984 AHSME (Problems • Answer Key • Resources)
Preceded by
Problem 25 		Followed by
Problem 27
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 • 26 • 27 • 28 • 29 • 30
All AHSME Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=1984_AHSME_Problems/Problem_26&oldid=83515"