Difference between revisions of "1963 AHSME Problems/Problem 12"

Latest revision as of 03:04, 7 June 2018

Problem

Three vertices of parallelogram $PQRS$ are $P(-3,-2), Q(1,-5), R(9,1)$ with $P$ and $R$ diagonally opposite. The sum of the coordinates of vertex $S$ is:

$\textbf{(A)}\ 13 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 11 \qquad \textbf{(D)}\ 10 \qquad \textbf{(E)}\ 9$

Solution

$[asy] import graph; size(7.22 cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-4.2,xmax=10.2,ymin=-6.2,ymax=5.2; pen cqcqcq=rgb(0.75,0.75,0.75), evevff=rgb(0.9,0.9,1), zzttqq=rgb(0.6,0.2,0); /*grid*/ pen gs=linewidth(0.7)+cqcqcq+linetype("2 2"); real gx=1,gy=1; for(real i=ceil(xmin/gx)*gx;i<=floor(xmax/gx)*gx;i+=gx) draw((i,ymin)--(i,ymax),gs); for(real i=ceil(ymin/gy)*gy;i<=floor(ymax/gy)*gy;i+=gy) draw((xmin,i)--(xmax,i),gs); Label laxis; laxis.p=fontsize(10); xaxis(xmin,xmax,defaultpen+black,Ticks(laxis,Step=1.0,Size=2,NoZero),Arrows(6),above=true); yaxis(ymin,ymax,defaultpen+black,Ticks(laxis,Step=1.0,Size=2,NoZero),Arrows(6),above=true); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); dot((-3,-2)); label("P",(-3,-2),NW); dot((1,-5)); label("Q",(1,-5),NW); dot((9,1)); label("R",(9,1),NW); [/asy]$ Graph the three points on the coordinate grid. Noting that the opposite sides of a parallelogram are congruent and parallel, count boxes to find out that point $S$ is on $(5,4)$ . The sum of the x-coordinates and y-coordinates is $9$ , so the answer is $\boxed{\textbf{(E)}}$ .

1963 AHSC (Problems • Answer Key • Resources)
Preceded by Problem 11	Followed by Problem 13
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
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The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

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Difference between revisions of "1963 AHSME Problems/Problem 12"

Latest revision as of 03:04, 7 June 2018

Problem

Solution

See Also

@@ Line 12: / Line 12: @@
 ==Solution==
 <asy>
-dot((-3,-2));
-label("P",(-3,-2),NW);
-dot((1,-5));
-label("Q",(1,-5),NW);
-dot((9,1));
-label("R",(9,1),NW);
 import graph; size(7.22 cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black;
-real xmin=-5.2,xmax=9.2,ymin=-5.2,ymax=6.2;
+real xmin=-4.2,xmax=10.2,ymin=-6.2,ymax=5.2;
 pen cqcqcq=rgb(0.75,0.75,0.75), evevff=rgb(0.9,0.9,1), zzttqq=rgb(0.6,0.2,0);
@@ Line 28: / Line 22: @@
 xaxis(xmin,xmax,defaultpen+black,Ticks(laxis,Step=1.0,Size=2,NoZero),Arrows(6),above=true); yaxis(ymin,ymax,defaultpen+black,Ticks(laxis,Step=1.0,Size=2,NoZero),Arrows(6),above=true);
 clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
+dot((-3,-2));
+label("P",(-3,-2),NW);
+dot((1,-5));
+label("Q",(1,-5),NW);
+dot((9,1));
+label("R",(9,1),NW);
 </asy>
 Graph the three points on the coordinate grid.  Noting that the opposite sides of a [[parallelogram]] are congruent and parallel, count boxes to find out that point <math>S</math> is on <math>(5,4)</math>.  The sum of the x-coordinates and y-coordinates is <math>9</math>, so the answer is <math>\boxed{\textbf{(E)}}</math>.
 ==See Also==