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.
Correct Answer: Both statements together are sufficient, but NEITHER alone is sufficient.
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.