Difference between revisions of "1950 AHSME Problems/Problem 48"
m (→Problem) |
|||
| (2 intermediate revisions by 2 users not shown) | |||
| Line 3: | Line 3: | ||
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: | 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: | ||
| − | <math>\textbf{(A)}\ \text{Least when the point is the center of gravity of the triangle}\qquad\\ | + | <math> \textbf{(A)}\ \text{Least when the point is the center of gravity of the triangle}\qquad\\ \textbf{(B)}\ \text{Greater than the altitude of the triangle}\qquad\\ \textbf{(C)}\ \text{Equal to the altitude of the triangle}\qquad\\ \textbf{(D)}\ \text{One-half the sum of the sides of the triangle}\qquad\\ \textbf{(E)}\ \text{Greatest when the point is the center of gravity} </math> |
| − | \textbf{(B)}\ \text{Greater than the altitude of the triangle} \qquad\\ | ||
| − | \textbf{(C)}\ \text{Equal to the altitude of the triangle}\qquad\\ | ||
| − | \textbf{(D)}\ \text{One-half the sum of the sides of the triangle} \qquad\\ | ||
| − | \textbf{(E)}\ \text{Greatest when the point is the center of gravity}</math> | ||
==Solution== | ==Solution== | ||
| Line 15: | Line 11: | ||
<cmath>PA'+PB'+PC'=h</cmath> | <cmath>PA'+PB'+PC'=h</cmath> | ||
The answer is <math>\textbf{(C)}</math> | The answer is <math>\textbf{(C)}</math> | ||
| + | ==Note== | ||
| + | Thie result is exactly the Viviani theorem. | ||
| + | |||
| + | |||
| + | ==Video Solution== | ||
| + | https://www.youtube.com/watch?v=l4lAvs2P_YA&t=251s | ||
| + | |||
| + | ~MathProblemSolvingSkills.com | ||
| + | |||
==See Also== | ==See Also== | ||
Latest revision as of 01:16, 2 November 2023
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
![\[\frac{1}{2}sPA'+\frac{1}{2}sPB'+\frac{1}{2}sPC'=\frac{1}{2}sh\]](http://latex.artofproblemsolving.com/f/7/6/f76759b8329b2e9dd1a58e22ad29072a74ac2215.png)
Where h is the altitude. Multiplying both sides by 
and dividing both sides by 
gives us
![\[PA'+PB'+PC'=h\]](http://latex.artofproblemsolving.com/6/e/a/6ea60170e4d145d40d38631dcc455414c64f2eab.png)
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 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50 | ||
| All AHSME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
