1967 AMC 12 Problem 13

Question

A triangle

ABC

is to be constructed given side

a

(opposite angle

A

), angle

B,

and

h_c,

the altitude from

C.

If

N

is the number of noncongruent solutions, then

N

is

1

is

2

must be zero

must be infinite

must be zero or infinite

Fix

BC=a

and draw the ray from

B

making the given angle

The distance from

C

to that ray is fixed, regardless of where

A

lies on it

Fix

BC=a

and draw the ray from

B

that makes the prescribed angle

B.

Its distance from

C

is fixed. If that distance is not

h_c,

there is no solution. If it is

h_c,

then

A

may be chosen at infinitely many positions on the ray, producing infinitely many noncongruent triangles. Thus

N

must be zero or infinite.

Therefore, the correct answer is E.

Problem 13 in other years