Difference between revisions of "1970 AHSME Problems/Problem 15"
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== Solution == | == Solution == | ||
| − | <math>\fbox{E}</math> | + | The trisection points of <math>(-4, 5)</math> and <math>(5, -1)</math> can be found by trisecting the x-coordinates and the y-coordinates separately. The difference of the x-coordinates is <math>9</math>, so the trisection points happen at <math>-4 + \frac{9}{3}</math> and <math>-4 + \frac{9}{3} + \frac{9}{3}</math>, which are <math>-1</math> and <math>2</math>. Similarly, the y-coordinates have a difference of <math>6</math>, so the trisections happen at <math>3</math> and <math>1</math>. So, the two points are <math>(-1, 3)</math> and <math>(2, 1)</math>. |
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| + | We now check which line has both <math>(3, 4)</math> and one of the two trisection points on it. Plugging in <math>(x, y) = (3, 4)</math> into all five of the equations works. The point <math>(2, 1)</math> doesn't work in any of the five lines. However, <math>(-1, 3)</math> works in line <math>E</math>. Thus, the answer is <math>\fbox{E}</math> | ||
== See also == | == See also == | ||
Latest revision as of 21:26, 13 July 2019
Problem
Lines in the 
-plane are drawn through the point 
and the trisection points of the line segment joining the points 
and 
. One of these lines has the equation
Solution
The trisection points of 
and 
can be found by trisecting the x-coordinates and the y-coordinates separately. The difference of the x-coordinates is 
, so the trisection points happen at 
and 
, which are 
and 
. Similarly, the y-coordinates have a difference of 
, so the trisections happen at 
and 
. So, the two points are 
and 
.
We now check which line has both 
and one of the two trisection points on it. Plugging in 
into all five of the equations works. The point doesn't work in any of the five lines. However, works in line 
. Thus, the answer is 
See also
| 1970 AHSC (Problems • Answer Key • Resources) | ||
| Preceded by Problem 14 |
Followed by Problem 16 | |
| 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 | ||
| All AHSME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
