1953 AMC 12 Problem 49

Question

The coordinates of

A,

B

and

C

are

(5,5),

(2,1)

and

(0,k)

respectively. The value of

k

that makes

\overline{AC}+\overline{BC}

as small as possible is:

3

4\dfrac12

3\dfrac67

4\dfrac56

2\dfrac17

Reflect

B

across the

y

-axis so that

BC

becomes the distance from

C

to the reflected point

The shortest broken path occurs where the straight line from

A

to the reflected point meets the

y

-axis

Reflect

B=(2,1)

across the

y

-axis to

B'=(-2,1).

For

C

on the

y

-axis,

BC=B'C,

so

AC+BC

is minimized when

A,C,B'

are collinear. The line from

A=(5,5)

to

B'=(-2,1)

has slope

\frac{4}{7}.

At

x=0,

its height is

5-\frac47(5)=\frac{15}{7}=2\frac17.

Hence

k=\frac{15}{7}.

Thus, the correct answer is E.

Problem 49 in other years