Show by the distance formula whether the points A(3, 0), B(6, 4) and C(−1, 3) form an isosceles right triangle. Pick the correct conclusion.

Class 10Coordinate GeometryGeometry — Coordinates

AYes, $A B = C A = 5$ and $A B^{2} + C A^{2} = B C^{2}$
BNo, all three sides are different
CYes, but it is equilateral
DNo, it is obtuse with no equal sides

Step-by-Step Solution

A B = (6 - 3)^{2} + (4 - 0)^{2} = 25 = 5. C A = (- 1 - 3)^{2} + (3 - 0)^{2} = 25 = 5. BC = (- 1 - 6)^{2} + (3 - 4)^{2} = 50 . A B = C A

(isosceles) and

A B^{2} + C A^{2} = 25 + 25 = 50 = B C^{2}

(right angle at A). So it is an isosceles right triangle.

Answer: A) Yes,

A B = C A = 5

and

A B^{2} + C A^{2} = B C^{2}

Key idea: Recognising shapes from coordinates

**Common tasks**:

- **Equilateral triangle**: all three sides equal length (use distance formula).
- **Isosceles**: two sides equal.
- **Right triangle**: side lengths satisfy Pythagoras

(a^{2} + b^{2} = c^{2})

.
- **Square**: all four sides equal AND adjacent sides perpendicular (or use diagonals: equal and bisect at right angles).
- **Rhombus**: all four sides equal but not necessarily right-angled.
- **Parallelogram**: opposite sides equal (use distance formula on all 4 sides — opposite pairs equal).
- **Rectangle**: opposite sides equal AND diagonals equal.

The procedure is always: compute side lengths via distance formula → match against the criteria for each shape.

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