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 $\boxed{\textbf{(D)}\}$ (Error compiling LaTeX. Unknown error_msg)
See Also
1964 AHSC (Problems • Answer Key • Resources) | ||
Preceded by Problem 16 |
Followed by Problem 18 | |
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