How far is point P from point Q? (All points lie on a straight line.) I. Point T is exactly midway between P and Q. Point T is 5 km west of point R. II. Point Q is 2 km east of point R.

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Solve this multiple-choice question and choose the correct option.

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Introduction / Context:We must compute the exact distance between two collinear points, P and Q, using reference points T and R. Sufficiency hinges on whether a unique numeric distance results. Given Data / Assumptions: I: T is the midpoint of segment PQ and T is 5 km west of R. II: Q is 2 km east of R. Positive eastward direction; west is negative relative to R for a simple coordinate model. Concept / Approach:Place R at coordinate 0. Use directional data to place T and Q, then apply the midpoint relation to solve for P and hence the distance PQ. Step-by-Step Solution: 1) Coordinate setup: Let R = 0. From II, Q = +2. From I, T = −5 and T is midpoint of P and Q ⇒ (P + Q)/2 = −5.2) Solve for P: (P + 2)/2 = −5 ⇒ P + 2 = −10 ⇒ P = −12.3) Distance PQ = |Q − P| = |2 − (−12)| = 14 km.4) Sufficiency: I alone lacks Q's absolute placement; II alone lacks the midpoint anchor. Together they yield a unique numerical distance. Verification / Alternative check:Reverse-check midpoint: (−12 + 2)/2 = −10/2 = −5, consistent with T's position. Why Other Options Are Wrong: A/B/C: Neither statement alone fixes unique coordinates; hence not sufficient individually. D: Together they are sufficient. Common Pitfalls:Sign errors with east/west; forgetting midpoint formula requires absolute positions, not just relative gaps. Final Answer:Both statements together are sufficient, but NEITHER alone is sufficient.

How far is point P from point Q? (All points lie on a straight line.) I. Point T is exactly midway between P and Q. Point T is 5 km west of point R. II. Point Q is 2 km east of point R.

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