Coordinate Geometry — Practice

Question 1

Find the distance between (3, 4) and (6, 8).

Accepted Answer

✦ Answer: d = √((6−3)² + (8−4)²) = √(9 + 16) = √25 = 5 units.

Question 2

Find the coordinates of the point dividing (2, 4) and (8, 10) in the ratio 2:1.

Accepted Answer

Step 1 — Apply section formula.
  x = (mx₂ + nx₁)/(m+n) = (2(8) + 1(2))/(2+1) = 18/3 = 6
  y = (my₂ + ny₁)/(m+n) = (2(10) + 1(4))/3 = 24/3 = 8

Step 2 — Verify.
  Distance from (2,4) to (6,8) = √(16+16) = √32
  Distance from (6,8) to (8,10) = √(4+4) = √8
  Ratio: √32 : √8 = 4√2 : 2√2 = 2:1 ✓

✦ Answer: (6, 8).

Question 3

Find the area of the triangle whose vertices are A(2,3), B(5,7), and C(8,3). Also check if it's right-angled.

Accepted Answer

Step 1 — Area formula.
  Area = (1/2)|x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)|
       = (1/2)|2(7−3) + 5(3−3) + 8(3−7)|
       = (1/2)|8 + 0 − 32|
       = (1/2)(24) = 12 sq units

Step 2 — Check sides.
  AB = √((5−2)² + (7−3)²) = √(9+16) = 5
  BC = √((8−5)² + (3−7)²) = √(9+16) = 5
  AC = √((8−2)² + (3−3)²) = √36 = 6

Step 3 — Check right angle.
  Check Pythagoras: 5² + 5² = 50; 6² = 36. Not equal, so NOT right-angled by AB-BC.
  Try AB-AC: 5² + 6² = 61. BC² = 25. Not equal.
  Try BC-AC: 5² + 6² = 61. AB² = 25. Not equal.
  NO right angle.

Step 4 — But isosceles!
  AB = BC = 5; AC = 6. Isosceles triangle.

✦ Answer: Area = 12 sq units. Triangle is ISOSCELES (AB = BC = 5) but NOT right-angled.

Practice — Coordinate Geometry

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