Question

In the xy-plane above, is angle QPR a right angle?
(1) Points P and Q have the same x-coordinate.
(2) Points P and R have the same y-coordinate.

• 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 D Each statement alone is sufficient.

Geometry Simple coordinate geometry

(1) Because points P and Q have the same x-coordinate, it follows that side $\overline{PQ}$ is vertical (i.e., parallel to the y-axis). Therefore, angle QPR will be a right angle if and only if side $\overline{PR}$ is horizontal (i.e.,parallel to the x-axis). However, depending on the location of point R in relation to point P, side $\overline{PR}$ may or may not be horizontal; NOT sufficient.

(2) Because points P and R have the samey-coordinate, it follows that side $\overline{PR}$ is horizontal. Therefore, angle QPR will be a right angle if and only if side $\overline{PQ}$ is vertical. However, depending on the location of point Q in relation to point P, side $\overline{PQ}$ mayor may not be vertical; NOT sufficient.

Given (1) and (2), it follows that $\overline{PQ}$ is vertical and $\overline{PR}$ is horizontal. Therefore, angle QPR is aright angle.

The correct answer is C; both statements together are sufficient.