1966 AHSME Problems/Problem 38
Problem
In triangle the medians and to sides and , respectively, intersect in point . is the midpoint of side , and intersects in . If the area of triangle is , then the area of triangle is:
Solution
Construct triangle with points being the midpoints of sides , respectively. Proceed by drawing all medians. Then draw all medians (so draw ). Next, draw line and label 's intersection with as the point . From the problem, the area of is , but by vertical angles we know that . Furthermore, since line is drawn from the midpoint of to the midpoint of , we know that is parallel to (via SAS similarity on triangles PCM and ABC). From these parallel lines we know that which indicates that . The linear ratio from to is 1:2 because line segment is one half of line segment since and make up the median . Thus the area ratio is 1:4. So has area . Since has the same height and base as we know that the area of . The medians form 3 triangles each with area of the total triangle (these triangles are ). Thus since .
- LJ
See also
1966 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 37 |
Followed by Problem 39 | |
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