1964 AHSME Problems/Problem 17
Problem 17
Given the distinct points 
and 
.
Line segments are drawn connecting these points to each other and to the origin 
.
Of the three possibilities: (1) parallelogram (2) straight line (3) trapezoid, figure 
,
depending upon the location of the points 
, and 
, can be:
Solution
Using vector addition can help solve this problem quickly. Note that algebraically, adding 
to 
will give 
. One method of vector addition is literally known as the "parallelogram rule" - if you are given and , to find , you can literally draw a parallelogram, making a line though 
parallel to 
, and a line through 
parallel to 
. The intersection of those lines will give the fourth point , and that fourth point will form a parallelogram with 
.
Thus, 
is a possibility. Case 
is also a possibility, if are collinear, then is also on that line.
Since 
and 
, which can be seen from either the prior reasoning or by examining slopes, the figure can never be a trapezoid, which requires exactly one of parallel sides.
Thus, the answer is 
See Also
| 1964 AHSC (Problems • Answer Key • Resources) | ||
| Preceded by Problem 16 |
Followed by Problem 18 | |
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| All AHSME Problems and Solutions | ||
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