# 1967 AHSME Problems/Problem 9

## Problem

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: $\textbf{(A)}\ K \; \text{must be an integer} \qquad \textbf{(B)}\ K \; \text{must be a rational fraction} \\ \textbf{(C)}\ K \; \text{must be an irrational number} \qquad \textbf{(D)}\ K \; \text{must be an integer or a rational fraction} \qquad$ $\textbf{(E)}\ \text{taken alone neither} \; \textbf{(A)} \; \text{nor} \; \textbf{(B)} \; \text{nor} \; \textbf{(C)} \; \text{nor} \; \textbf{(D)} \; \text{is true}$

## Solution

From the problem we can set the altitude equal to $a$, the shorter base equal to $a-d$, and the longer base equal to $a+d$. By the formula for the area of a trapezoid, we have $K=a^2$. However, since $a$ can equal any real number $(3, 2.7, \pi)$, none of the statements $A, B, C, D$ need to be true, so the answer is $\fbox{E}$.

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