Difference between revisions of "1967 AHSME Problems/Problem 37"
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== Solution == | == Solution == | ||
<math>\fbox{A}</math> | <math>\fbox{A}</math> | ||
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| + | 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 == | ||
| − | {{AHSME box|year=1967|num-b=36|num-a=38}} | + | {{AHSME 40p box|year=1967|num-b=36|num-a=38}} |
[[Category: Intermediate Geometry Problems]] | [[Category: Intermediate Geometry Problems]] | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Latest revision as of 00:40, 16 August 2023
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 AHSC (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 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 | ||
| All AHSME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
