Difference between revisions of "2007 AMC 12B Problems/Problem 14"
Chickendude (talk | contribs) (New page: ==Problem 14== Point <math>P</math> is inside equilateral <math>\triangle ABC</math>. Points <math>Q</math>, <math>R</math>, and <math>S</math> are the feet of the perpendiculars from <mat...) |
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Point <math>P</math> is inside equilateral <math>\triangle ABC</math>. Points <math>Q</math>, <math>R</math>, and <math>S</math> are the feet of the perpendiculars from <math>P</math> to <math>\overline{AB}</math>, <math>\overline{BC}</math>, and <math>\overline{CA}</math>, respectively. Given that <math>PQ=1</math>, <math>PR=2</math>, and <math>PS=3</math>, what is <math>AB</math>? | Point <math>P</math> is inside equilateral <math>\triangle ABC</math>. Points <math>Q</math>, <math>R</math>, and <math>S</math> are the feet of the perpendiculars from <math>P</math> to <math>\overline{AB}</math>, <math>\overline{BC}</math>, and <math>\overline{CA}</math>, respectively. Given that <math>PQ=1</math>, <math>PR=2</math>, and <math>PS=3</math>, what is <math>AB</math>? | ||
− | <math>\mathrm {(A)} 4 | + | <math>\mathrm{(A)}\ 4 \qquad \mathrm{(B)}\ 3\sqrt{3} \qquad \mathrm{(C)}\ 6 \qquad \mathrm{(D)}\ 4\sqrt{3} \qquad \mathrm{(E)}\ 9</math> |
==Solution== | ==Solution== |
Revision as of 23:07, 20 February 2008
Problem 14
Point is inside equilateral . Points , , and are the feet of the perpendiculars from to , , and , respectively. Given that , , and , what is ?
Solution
Drawing , , and , is split into three smaller triangles. The altitudes of these triangles are given in the problem as , , and .
Summing the areas of each of these triangles and equating it to the area of the entire triangle, we get:
where is the length of a side