Difference between revisions of "2001 AMC 12 Problems/Problem 20"

Revision as of 07:03, 9 May 2011

Problem

Points $A = (3,9)$ , $B = (1,1)$ , $C = (5,3)$ , and $D=(a,b)$ lie in the first quadrant and are the vertices of quadrilateral $ABCD$ . The quadrilateral formed by joining the midpoints of $\overline{AB}$ , $\overline{BC}$ , $\overline{CD}$ , and $\overline{DA}$ is a square. What is the sum of the coordinates of point $D$ ?

$\text{(A) }7 \qquad \text{(B) }9 \qquad \text{(C) }10 \qquad \text{(D) }12 \qquad \text{(E) }16$

Solution

$[asy] pair A=(3,9), B=(1,1), C=(5,3), D=(7,3); draw(A--B--C--D--cycle); label("$A$",A,N); label("$B$",B,SW); label("$C$",C,N); label("$D$",D,E); pair AB = (A + B)/2, BC = (B + C)/2, CD = (C + D)/2, DA = (D + A)/2; draw(AB--BC--CD--DA--cycle); [/asy]$

We already know two vertices of the square: $(A+B)/2 = (2,5)$ and $(B+C)/2 = (3,2)$ .

There are only two possibilities for the other vertices of the square: either they are $(6,3)$ and $(5,6)$ , or they are $(0,1)$ and $(-1,4)$ . The second case would give us $D$ outside the first quadrant, hence the first case is the correct one. As $(6,3)$ is the midpoint of $CD$ , we can compute $D=(7,3)$ , and $7+3=\boxed{10}$ .

2001 AMC 12 (Problems • Answer Key • Resources)
Preceded by Problem 19	Followed by Problem 21
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
All AMC 12 Problems and Solutions

Page

Toolbox

Search

Difference between revisions of "2001 AMC 12 Problems/Problem 20"

Revision as of 07:03, 9 May 2011

Problem

Solution

See Also

@@ Line 22: / Line 22: @@
 label("$A$",A,N);
 label("$B$",B,SW);
-label("$C$",C,S);
+label("$C$",C,N);
 label("$D$",D,E);
+pair AB = (A + B)/2, BC = (B + C)/2, CD = (C + D)/2, DA = (D + A)/2;
+draw(AB--BC--CD--DA--cycle);
 </asy>