Difference between revisions of "1987 AHSME Problems/Problem 14"
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\textbf{(C)}\ \frac{\sqrt{10}}{5} \qquad | \textbf{(C)}\ \frac{\sqrt{10}}{5} \qquad | ||
\textbf{(D)}\ \frac{4}{5}\qquad | \textbf{(D)}\ \frac{4}{5}\qquad | ||
| − | \textbf{(E)}\ \text{none of these} </math> | + | \textbf{(E)}\ \text{none of these} </math> |
| + | ==Solution== | ||
| + | Use the Sine Area Formula. We can isolate the triangle for which the angle <math>\theta</math> is contained in. WLOG, denote the side length of a triangle as <math>2</math>. Our midpoints are then <math>1</math>. Subtract the areas of the triangles that don't include the area of our desired triangle: <math>4-1-1-\frac{1}{2} = \frac{3}{2}.</math> The Sine Area Formula tells us <math>\frac{1}{2}(\sqrt{5})^2\sin\theta=\frac{3}{2}.</math> Solving this equation, we get <math>\sin\theta=\textbf{(B)}\ \frac{3}{5} \qquad</math> | ||
| + | Solution: Everyoneintexas | ||
== See also == | == See also == | ||
Latest revision as of 23:32, 14 February 2018
Problem
is a square and 
and 
are the midpoints of 
and 
respectively. Then 
![[asy] draw((0,0)--(2,0)--(2,2)--(0,2)--cycle); draw((0,0)--(2,1)); draw((0,0)--(1,2)); label("A", (0,0), SW); label("B", (0,2), NW); label("C", (2,2), NE); label("D", (2,0), SE); label("M", (1,2), N); label("N", (2,1), E); label("$\theta$", (.5,.5), SW); [/asy]](http://latex.artofproblemsolving.com/5/4/9/5492ba158043d653ed4090a75df79115e58fdb4f.png)
Solution
Use the Sine Area Formula. We can isolate the triangle for which the angle 
is contained in. WLOG, denote the side length of a triangle as 
. Our midpoints are then 
. Subtract the areas of the triangles that don't include the area of our desired triangle: 
The Sine Area Formula tells us 
Solving this equation, we get 
Solution: Everyoneintexas
See also
| 1987 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 13 |
Followed by Problem 15 | |
| 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. 
