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 "1986 AJHSME Problems/Problem 13"
m
Line 15: Line 15:
 
==Solution==
 
==Solution==
  
−
Note: I'm assuming all concave angles are right angles.
+
For the segments parallel to the side with side length 8, let's call those two segments <math>a</math> and <math>b</math>, the longer segment being <math>b</math>, the shorter one being <math>a</math>.
  
−
At first, we may immediately put down E, because of course we don't know many of the segments. But let's give it a shot.
+
For the segments parallel to the side with side length 6, let's call those two segments <math>c</math> and <math>d</math>, the longer segment being <math>d</math>, the shorter one being <math>c</math>.
−
 
 
−
For the segments parallel to the side with side length 8, let's call those two segments <math>a</math> and <math>b</math>, the longer segment being b, the shorter one being a.
 
−
 
 
−
For the segments parallel to the side with side length 6, let's call those two segments <math>c</math> and <math>d</math>, the longer segment being d, the shorter one being c.
 
  
 
So the perimeter of the polygon would be...
 
So the perimeter of the polygon would be...
Line 27: Line 23:
 
<math>8 + 6 + a + b + c + d</math>
 
<math>8 + 6 + a + b + c + d</math>
  
−
And it looks like we're stuck. But wait!
+
Note that <math>a + b = 8</math>, and <math>c + d = 6</math>.
−
 
 
−
Notice that <math>a + b = 8</math>, and <math>c + d = 6</math>.
 
−
 
 
−
Now we plug those in...
 
  
−
<math>8 + 6 + a + b + c + d = 8 + 6 + 8 + 6 = 14 \times 2 = 28</math>
+
Now we plug those in:
 +
<math></math>\begin{align*}
 +
8 + 6 + a + b + c + d &= 8 + 6 + 8 + 6 \\
 +
&= 14 \times 2 \\
 +
&= 28 \\
 +
\end{align*}<math>
  
−
28 is C.
+
28 is </math>\boxed{\text{C}}$.
  
 
==See Also==
 
==See Also==
  
 
[[1986 AJHSME Problems]]
 
[[1986 AJHSME Problems]]

Revision as of 19:24, 24 January 2009

Problem

The perimeter of the polygon shown is

[asy] draw((0,0)--(0,6)--(8,6)--(8,3)--(2.7,3)--(2.7,0)--cycle); label("$6$",(0,3),W); label("$8$",(4,6),N); [/asy]

$\text{(A)}\ 14 \qquad \text{(B)}\ 20 \qquad \text{(C)}\ 28 \qquad \text{(D)}\ 48$

$\text{(E)}\ \text{cannot be determined from the information given}$

Solution

For the segments parallel to the side with side length 8, let's call those two segments $a$ and $b$, the longer segment being $b$, the shorter one being $a$.

For the segments parallel to the side with side length 6, let's call those two segments $c$ and $d$, the longer segment being $d$, the shorter one being $c$.

So the perimeter of the polygon would be...

$8 + 6 + a + b + c + d$

Note that $a + b = 8$, and $c + d = 6$.

Now we plug those in: $$ (Error compiling LaTeX. Unknown error_msg)\begin{align*} 8 + 6 + a + b + c + d &= 8 + 6 + 8 + 6 \\ &= 14 \times 2 \\ &= 28 \\ \end{align*}$28 is$\boxed{\text{C}}$.

See Also

1986 AJHSME Problems

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=1986_AJHSME_Problems/Problem_13&oldid=29723"