Difference between revisions of "University of South Carolina High School Math Contest/1993 Exam/Problem 16"
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== Problem == | == Problem == | ||
In the triangle below, <math>\displaystyle M, N, </math> and <math>P</math> are the midpoints of <math>BC, AB,</math> and <math>AC</math> respectively. <math>CN</math> and <math>AM</math> intersect at <math>O</math>. If the length of <math>CQ</math> is 4, then what is the length of <math>OQ</math>? | In the triangle below, <math>\displaystyle M, N, </math> and <math>P</math> are the midpoints of <math>BC, AB,</math> and <math>AC</math> respectively. <math>CN</math> and <math>AM</math> intersect at <math>O</math>. If the length of <math>CQ</math> is 4, then what is the length of <math>OQ</math>? | ||
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+ | <center>[[Image:Usc93.16.PNG]]</center> | ||
<center><math> \mathrm{(A) \ }1 \qquad \mathrm{(B) \ }4/3 \qquad \mathrm{(C) \ }\sqrt{2} \qquad \mathrm{(D) \ }3/2 \qquad \mathrm{(E) \ }2 </math></center> | <center><math> \mathrm{(A) \ }1 \qquad \mathrm{(B) \ }4/3 \qquad \mathrm{(C) \ }\sqrt{2} \qquad \mathrm{(D) \ }3/2 \qquad \mathrm{(E) \ }2 </math></center> | ||
== Solution == | == Solution == | ||
+ | <math>AM</math> and <math>CN</math> are the [[median of a triangle|medians]] of <math>\triangle ABC</math>, so their intersection point <math>O</math> is the [[centroid]] of the [[triangle]]. Also, <math>\frac{CM}{MB} = \frac{CP}{PA} = 1</math> so <math>MP</math> is [[parallel]] to <math>AB</math> and thus <math>\frac{CQ}{QN} = 1</math> and <math>CQ = QN = 4</math>. Then <math>CN = CQ + QN = 8</math>. Since the centroid trisects the medians, <math>CO = \frac23 CN = \frac{16}3</math> and <math>OQ = CO - CQ = \frac{16}3 - 4 = \frac43</math> which is answer choice <math>\mathrm{(B)}</math>. | ||
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Latest revision as of 11:45, 17 August 2006
Problem
In the triangle below, and
are the midpoints of
and
respectively.
and
intersect at
. If the length of
is 4, then what is the length of
?
![$\mathrm{(A) \ }1 \qquad \mathrm{(B) \ }4/3 \qquad \mathrm{(C) \ }\sqrt{2} \qquad \mathrm{(D) \ }3/2 \qquad \mathrm{(E) \ }2$](http://latex.artofproblemsolving.com/3/b/4/3b4b0d0b2d5665b32ea30490e6d82ef9402ece4e.png)
Solution
and
are the medians of
, so their intersection point
is the centroid of the triangle. Also,
so
is parallel to
and thus
and
. Then
. Since the centroid trisects the medians,
and
which is answer choice
.