Question

In the figure above, is the area of triangular region ABC equal to the area of triangular region DBA?
(1) $(AC{)}^2=2(AD{)}^2$
(2) $△ABC$ is isosceles.

• A. Statement (1) alone is sufficient, but statement (2) alone is not sufficient.
• B. Statement (2) alone is sufficient, but statement (1) alone is not sufficient.
• C. Both statements together are sufficient, but neither statement alone is sufficient.
• D. Each statement alone is sufficient.
• E. Statements (1) and (2) together are not sufficient.

Answer 1

The correct option is C Both statements together are sufficient, but neither statement alone is sufficient.

• Determine whether or, equivalently, whether (AC)(CB) = (AD)(AB).

1. Given that , then and 1 could be the values of AC and AD, respectively. If, for example, CB= , then by the Pythagorean theorem applied to ∆ABC. Hence, and (AD)(AB) = (1)(2), from which it follows that (AC)(CB) = (AD)(AB). On the other hand, if CB = 1, then by the Pythagorean theorem applied to ∆ABC. Hence, and , from which it follows that (AC)(CB) ≠ (AD)(AB); NOT sufficient.
2. Given that ∆ABC is isosceles then, by varying the value of AD, the area of ∆ABC may or may not be equal to the area of ∆DBA. For example, if AC = CB = 1, then (AC)(CB) = (1)(1) = 1 and AB = by the Pythagorean theorem applied to ∆ABC. If then and (AC)(CB) = (AD)(AB). On the other hand, if AD = 1, and (AC)(CB) ≠ (AD)(AB); NOT sufficient.

• Given (1) and (2) together, let AC = CB = x be the length of the legs of the isosceles triangle ABC. Then, from , it follows that , and hence . Also, by the Pythagorean theorem applied to ∆ABC, it follows that . Therefore, it follows that (AC)(CB) = (x)(x) = and , and so (AC)(CB) = (AD)(AB).
• The correct answer is C; both statements together are sufficient.