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A student claims that A(1, 2), B(5, 2), C(6, 5) and D(2, 5) form a rectangle, because AB = CD and BC = AD. Is the claim correct?

Class 10Coordinate GeometryGeometry — Coordinates
  • AYes — equal opposite sides is enough to prove a rectangle
  • BNo — the four points form a rhombus instead
  • CYes — and it is in fact a square
  • DNo — the diagonals AC = and BD = are unequal, so it is only a parallelogram

Step-by-Step Solution

AB = 4, CD = , BC = , AD = , so the opposite sides really are equal and ABCD is a parallelogram. But a rectangle also needs EQUAL DIAGONALS, and AC = while BD = — unequal, so there is no right angle and the claim fails. It is not a rhombus either, since 4 ≠ .
Answer: D) No — the diagonals AC = and BD = are unequal, so it is only a parallelogram

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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