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Difference between revisions of "1964 AHSME Problems/Problem 17"

(Created page with "== Problem 17== Given the distinct points <math>P(x_1, y_1), Q(x_2, y_2)</math> and <math>R(x_1+x_2, y_1+y_2)</math>. Line segments are drawn connecting these points to each...")
 
(→‎Solution)
 
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\textbf{(E)}\ \text{all three} </math>   
 
\textbf{(E)}\ \text{all three} </math>   
  
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==Solution =
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==Solution==
  
 
Using vector addition can help solve this problem quickly.  Note that algebraically, adding <math>\overrightarrow{OP}</math> to <math>\overrightarrow{OQ}</math> will give <math>\overrightarrow{OR}</math>.  One method of vector addition is literally known as the "parallelogram rule" - if you are given <math>\overrightarrow{OP}</math> and <math>\overrightarrow{OQ}</math>, to find <math>\overrightarrow{OR}</math>, you can literally draw a parallelogram, making a line though <math>P</math> parallel to <math>OQ</math>, and a line through <math>Q</math> parallel to <math>OP</math>.  The intersection of those lines will give the fourth point <math>R</math>, and that fourth point will form a parallelogram with <math>O, P, Q</math>.
 
Using vector addition can help solve this problem quickly.  Note that algebraically, adding <math>\overrightarrow{OP}</math> to <math>\overrightarrow{OQ}</math> will give <math>\overrightarrow{OR}</math>.  One method of vector addition is literally known as the "parallelogram rule" - if you are given <math>\overrightarrow{OP}</math> and <math>\overrightarrow{OQ}</math>, to find <math>\overrightarrow{OR}</math>, you can literally draw a parallelogram, making a line though <math>P</math> parallel to <math>OQ</math>, and a line through <math>Q</math> parallel to <math>OP</math>.  The intersection of those lines will give the fourth point <math>R</math>, and that fourth point will form a parallelogram with <math>O, P, Q</math>.
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Since <math>OP \parallel QR</math> and <math>PQ \parallel RO</math>, 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.
 
Since <math>OP \parallel QR</math> and <math>PQ \parallel RO</math>, 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.
  
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Thus, the answer is <math>\boxed{\textbf{(D)}\}</math>
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Thus, the answer is <math>\boxed{\textbf{(D)}}</math>
  
 
==See Also==
 
==See Also==

Latest revision as of 15:10, 5 July 2021

Problem 17

Given the distinct points $P(x_1, y_1), Q(x_2, y_2)$ and $R(x_1+x_2, y_1+y_2)$. Line segments are drawn connecting these points to each other and to the origin $O$. Of the three possibilities: (1) parallelogram (2) straight line (3) trapezoid, figure $OPRQ$, depending upon the location of the points $P, Q$, and $R$, can be:

$\textbf{(A)}\ \text{(1) only}\qquad \textbf{(B)}\ \text{(2) only}\qquad \textbf{(C)}\ \text{(3) only}\qquad \textbf{(D)}\ \text{(1) or (2) only}\qquad \textbf{(E)}\ \text{all three}$

Solution

Using vector addition can help solve this problem quickly. Note that algebraically, adding $\overrightarrow{OP}$ to $\overrightarrow{OQ}$ will give $\overrightarrow{OR}$. One method of vector addition is literally known as the "parallelogram rule" - if you are given $\overrightarrow{OP}$ and $\overrightarrow{OQ}$, to find $\overrightarrow{OR}$, you can literally draw a parallelogram, making a line though $P$ parallel to $OQ$, and a line through $Q$ parallel to $OP$. The intersection of those lines will give the fourth point $R$, and that fourth point will form a parallelogram with $O, P, Q$.

Thus, $1$ is a possibility. Case $2$ is also a possibility, if $O, P, Q$ are collinear, then $R$ is also on that line.

Since $OP \parallel QR$ and $PQ \parallel RO$, 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 $\boxed{\textbf{(D)}}$

See Also

1964 AHSC (Problems • Answer Key • Resources)
Preceded by
Problem 16 		Followed by
Problem 18
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

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