1979 AMC 12 Problem 17

Question

Points

A,

B,

C,

and

D

are distinct and lie, in the given order, on a straight line. Line segments

AB,

AC,

and

AD

have lengths

x,

y,

and

z,

respectively. If line segments

AB

and

CD

may be rotated about points

B

and

C,

respectively, so that points

A

and

D

coincide, to form a triangle with positive area, then which of the following three inequalities must be satisfied?

\begin{aligned} &\text{I. }x\lt\frac z2\\ &\text{II. }y\lt x+\frac z2\\ &\text{III. }y\lt\frac z2 \end{aligned}

\mathrm{I}

only

\mathrm{II}

only

\mathrm{I}

and

\mathrm{II}

only

\mathrm{II}

and

\mathrm{III}

only

\mathrm{I},

\mathrm{II}

and

\mathrm{III}

After the rotations, identify the three side lengths of the new triangle

Apply all three strict triangle inequalities to

x,

y-x,

and

z-y

The new triangle has side lengths

x,

y-x,

and

z-y.

Its triangle inequalities give

\begin{aligned} x&\lt(y-x)+(z-y),\\ &\Longrightarrow x\lt \frac{z}{2};\\ y-x&\lt x+(z-y),\\ &\Longrightarrow y\lt x+\frac{z}{2};\\ z-y&\lt x+(y-x),\\ &\Longrightarrow y\gt \frac{z}{2}. \end{aligned}

Thus I and II must hold, while III must fail.

Therefore, the correct answer is C.

Problem 17 in other years