2003 AMC 8 Problems/Problem 24

Problem

A ship travels from point $A$ to point $B$ along a semicircular path, centered at Island $X$ . Then it travels along a straight path from $B$ to $C$ . Which of these graphs best shows the ship's distance from Island $X$ as it moves along its course?

$[asy]size(150); pair X=origin, A=(-5,0), B=(5,0), C=(0,5); draw(Arc(X, 5, 180, 360)^^B--C); dot(X); label("$X$", X, NE); label("$C$", C, N); label("$B$", B, E); label("$A$", A, W); [/asy]$

Solution

The distance from $\text{X}$ to any point on the semicircle will always be constant. On the graph, this will represent a straight line. The distance between $\text{X}$ and line $\text{BC}$ will not be constant though. We can easily prove that the distance between $\text{X}$ and line $\text{BC}$ will represent a semicircle (prove this by dividing $\triangle{XCB}$ into two congruent triangles using the perpendicular bisector from vertex $\text{X}$ ). Since the point on line $\text{BC}$ and the perpendicular bisector from vertex $\text{X}$ is the shortest distance between $\text{X}$ and $\text{BC}$ as well as the midpoint of line $\text{BC}$ it will represent the shortest point on the semicircle in the graph as well as the midpoint of the semicircle. Using the information found, the answer choice that fits them all is $\boxed{\text{(B)}}$ .

2003 AMC 8 (Problems • Answer Key • Resources)
Preceded by Problem 23	Followed by Problem 25
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 AJHSME/AMC 8 Problems and Solutions

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2003 AMC 8 Problems/Problem 24

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