1950 AHSME Problems/Problem 48
Contents
Problem
A point is selected at random inside an equilateral triangle. From this point perpendiculars are dropped to each side. The sum of these perpendiculars is:
Solution
Begin by drawing the triangle, the point, the altitudes from the point to the sides, and the segments connecting the point to the vertices. Let the triangle be with . We will call the aforementioned point . Call altitude from to . Similarly, we will name the other two altitudes and . We can see that Where h is the altitude. Multiplying both sides by and dividing both sides by gives us The answer is
Note
Thie result is exactly the Viviani theorem.
Video Solution
https://www.youtube.com/watch?v=l4lAvs2P_YA&t=251s
~MathProblemSolvingSkills.com
See Also
1950 AHSC (Problems • Answer Key • Resources) | ||
Preceded by Problem 47 |
Followed by Problem 49 | |
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