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Using distances only, decide whether P(−2, −1), Q(1, 3) and R(4, 7) lie on one straight line.

Class 10Coordinate GeometryGeometry — Coordinates
  • ANo — PR is longer than PQ + QR
  • BYes — because PQ = QR
  • CYes — PQ + QR = PR, so the three points are collinear
  • DNo — PQ = QR only makes triangle PQR isosceles

Step-by-Step Solution

PQ = ; QR = ; PR = . Since PQ + QR = 5 + 5 = 10 = PR, Q lies on the segment PR and the points are collinear. Note that PQ = QR on its own would prove nothing — equal distances alone only give an isosceles triangle; it is the SUM test that settles collinearity.
Answer: C) Yes — PQ + QR = PR, so the three points are collinear

Key idea: Distance formula

**Distance** between two points P₁, ₁) and Q₂, ₂):

**PQ = **

This is just the Pythagoras theorem applied to the right triangle formed with horizontal and vertical legs.

**Example**: Distance between (3, 4) and (7, 1).
= .

**Special distance from origin**: distance of from (0, 0) is .

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