Difference between revisions of "1967 AHSME Problems/Problem 37"
(Created page with "== Problem == Segments <math>AD=10</math>, <math>BE=6</math>, <math>CF=24</math> are drawn from the vertices of triangle <math>ABC</math>, each perpendicular to a straight line <...") |
Atmchallenge (talk | contribs) (→Solution) |
||
Line 10: | Line 10: | ||
== Solution == | == Solution == | ||
<math>\fbox{A}</math> | <math>\fbox{A}</math> | ||
+ | |||
+ | WLOG let <math>RS</math> be the x-axis, or at least horizontal. The three lengths represent the y-coordinates of points <math>A,B,C</math>. As <math>G</math> is by definition the average of <math>A,B,C</math>, it's coordinates are the average of the coordinates of <math>A,B,</math> and <math>C</math>. Hence, the y-coordinate of <math>G</math>, which is also the distance from <math>G</math> to <math>RS</math>, is the average of the y-coordinates of <math>A,B,</math> and <math>C</math>, or <math>\frac{10+6+24}{3}</math>, hence the result. | ||
== See also == | == See also == |
Latest revision as of 01:04, 8 June 2018
Problem
Segments , , are drawn from the vertices of triangle , each perpendicular to a straight line , not intersecting the triangle. Points , , are the intersection points of with the perpendiculars. If is the length of the perpendicular segment drawn to from the intersection point of the medians of the triangle, then is:
Solution
WLOG let be the x-axis, or at least horizontal. The three lengths represent the y-coordinates of points . As is by definition the average of , it's coordinates are the average of the coordinates of and . Hence, the y-coordinate of , which is also the distance from to , is the average of the y-coordinates of and , or , hence the result.
See also
1967 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 36 |
Followed by Problem 38 | |
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 | ||
All AHSME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.