1967 AMC 12 Problem 9

Question

Let

K,

in square units, be the area of a trapezoid such that the shorter base, the altitude, and the longer base, in that order, are in arithmetic progression. Then:

K

must be an integer

K

must be a rational fraction

K

must be an irrational number

K

must be an integer or a rational fraction

taken alone neither

(A)

nor

(B)

nor

(C)

nor

(D)

is true

Answer: E

Concepts:trapezoid arithmetic sequence area

Difficulty rating: 1650

Small hint:

Write the three lengths as

a-d,

a,

and

a+d

Big hint:

The resulting area is

a^2,

but the problem does not restrict the kind of number

a

is

Solution:

Let the shorter base, altitude, and longer base be

a-d,

a,

and

a+d.

Then

\begin{aligned} K&=\frac12a\bigl((a-d)+(a+d)\bigr)\\ &=a^2. \end{aligned}

Depending on

a,

this can be an integer, a nonintegral rational number, or an irrational number. No one of choices (A) through (D) must hold.

Therefore, the correct answer is E.