Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 2.1 Junior Find the answer

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

Your answer

Multiple choice: answer with the letter of the option you want.

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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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.