2011 AMC 10B Problems/Problem 18
Contents
[hide]Problem
Rectangle has
and
. Point
is chosen on side
so that
. What is the degree measure of
?
Solution 1
![[asy] unitsize(10mm); defaultpen(linewidth(.5pt)+fontsize(10pt)); dotfactor=3; pair A=(0,3), B=(6,3), C=(6,0), D=(0,0); pair M=(0.80385,3); draw(A--B--C--D--cycle); draw(M--C); draw(M--D); draw(anglemark(A,M,D)); draw(anglemark(D,M,C)); draw(anglemark(C,D,M)); pair[] ps={A,B,C,D,M}; dot(ps); label("$A$",A,NW); label("$B$",B,NE); label("$C$",C,SE); label("$D$",D,SW); label("$M$",M,N); label("$6$",midpoint(C--M),SW); label("$3$",midpoint(B--C),E); label("$6$",midpoint(C--D),S); [/asy]](http://latex.artofproblemsolving.com/a/3/6/a36bfc01ce63b46fe1eb4a5e9b245fbd9903e9f5.png)
It is given that . Since
and
are alternate interior angles and
,
. Use the Base Angle Theorem to show
. We know that
is a rectangle, so it follows that
. We notice that
is a
triangle, and
. If we let
be the measure of
then
Solution 2 (with trig)
Let . If we let
, we have that
, by the Pythagorean Theorem, and similarily,
. Applying the law of cosine, we see that
and
YAY!!! We have two equations for two variables... that are terribly ugly. Well, we'll try to solve it. First of all, note that
, so solving for
in terms of
, we get that
. The equation now becomes
Simplifying, we get
Now, we apply the quartic formula to get
We can easily see that is an invalid solution. Thus,
.
Finally, since ,
, where
is any integer. Converting to degrees, we have that
. Since
, we have that
.
~ilovepi3.14
Solution 3(Easier Trig)
We have . By Pythag,
, and thus
, We have
, or angle AM=
~awsomek
See Also
2011 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 17 |
Followed by Problem 19 | |
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 | ||
All AMC 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.