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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