1999 AHSME Problems/Problem 22

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Problem

The graphs of $y = -|x-a| + b$ and $y = |x-c| + d$ intersect at points $(2,5)$ and $(8,3)$ . Find $a+c$ .

$\mathrm{(A) \ } 7 \qquad \mathrm{(B) \ } 8 \qquad \mathrm{(C) \ } 10 \qquad \mathrm{(D) \ } 13\qquad \mathrm{(E) \ } 18$

Solution

Each of the graphs consists of two orthogonal half-lines. In the first graph both point downwards at a $45^\circ$ angle, in the second graph they point upwards. One can easily find out that the only way how to get these graphs to intersect in two points is the one depicted below:

unitsize(0.5cm);pair X=(2,5), Y=(8,3);draw ( (-1,2) -- (4,7) -- (10,1) );draw ( (-1,8) -- (6,1) -- (10,5) );label("$(2,5...

Obviously, the maximum of the first graph is achieved when $x=a$ , and its value is $-0+b=b$ . Similarly, the minimum of the other graph is $(c,d)$ . Therefore the two remaining vertices of the area between the graphs are $(a,b)$ and $(c,d)$ .

As the area has four right angles, it is a rectangle. Without actually computing $a$ and $c$ we can therefore conclude that $a+c=2+8=\boxed{10}$ .

Explanation of the last step

This is a property all rectangles in the coordinate plane have.

For a proof, note that for any rectangle $ABCD$ its center can be computed as $(A+C)/2$ and at the same time as $(B+D)/2$ . In our case, we can compute that the center is $\left(\frac{2+8}2,\frac{5+3}2\right)=(5,4)$ , therefore $\frac{a+c}2=5$ , and $a+c=10$ .

unitsize(0.5cm);pair X=(2,5), Y=(8,3);draw ( (-1,2) -- (4,7) -- (10,1) );draw ( (-1,8) -- (6,1) -- (10,5) );draw ( (4,7) -- (...

An alternate last step

We can easily compute $a$ and $c$ using our picture.

Consider the first graph on the interval $[2,8]$ . The graph starts at height $5$ , then rises for $a-2$ steps to the height $b=5+(a-2)$ , and then falls for $8-a$ steps to the height $3=5+(a-2)-(8-a)$ . Solving for $a$ we get $a=4$ . Similarly we compute $c=6$ , therefore $a+c=10$ .

1999 AHSME (Problems)
Preceded by Problem 21	Followed by Problem 23
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

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