1970 AMC 12 Problem 20

Attempt Problem 20 of the 1970 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1970 AMC 12 solutions, or check the answer key.

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20.

Lines

HK

and

BC

lie in a plane.

M

is the midpoint of line segment

BC,

and

BH

and

CK

are perpendicular to

HK.

Then we:

always have

MH=MK

always have

MH\gt BK

sometimes have

MH=MK

but not always

always have

MH\gt MB

always have

BH\lt BC

Put

HK

on the

x

-axis

Use

H=(h,0),

B=(h,u),

K=(k,0),

and

C=(k,v)

; compare squared distances from their midpoint

Choose coordinates

\begin{gathered} H=(h,0),\quad B=(h,u),\\ K=(k,0),\quad C=(k,v). \end{gathered}

Then

M=\left(\frac{h+k}{2},\frac{u+v}{2}\right).

The horizontal displacements from

M

H

and

K

are opposites, while the vertical displacements are equal. Therefore

MH^2=MK^2,

MH=MK

always.

Therefore, the correct answer is A.

Problem 20 in other years