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>? | ||
− | <center><math> \mathrm{(A) \ } \qquad \mathrm{(B) \ } \qquad \mathrm{(C) \ } \qquad \mathrm{(D) \ } \qquad \mathrm{(E) \ } </math></center> | + | <center>[[Image:Usc93.16.PNG]]</center> |
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+ | <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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+ | * [[University of South Carolina High School Math Contest/1993 Exam/Problem 15|Previous Problem]] | ||
+ | * [[University of South Carolina High School Math Contest/1993 Exam/Problem 17|Next Problem]] | ||
+ | * [[University of South Carolina High School Math Contest/1993 Exam|Back to Exam]] | ||
− | + | [[Category:Intermediate Geometry Problems]] | |
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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 ?
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 .