# Worksheet on Co-ordinate Triangle

In the worksheet on co-ordinate triangle we need to find the area of a triangle where the three co-ordinates of the vertices are given.

Let us recall the formula for finding the area of a triangle formed by joining the three given points as follows;

In terms of Cartesian co-ordinates the area of a triangle formed by joining the points (x₁, y₁), (x₂, y₂) and (x₃, y₃) is

½ | y₁ (x₂ - x₃) + y₂ (x₃ - x₁) + y₃ (x₁ - x₂) | sq. units

or, ½ | x₁ (y₂ - y₃) + x₂ (y₃ - y₁) + x₃ (y₁ - y₂) | sq. units.

In terms of polar co-ordinates (x₁, y₁), (x₂, y₂) and (x₃, y₃) of the vertices A, B, C respectively.

∆ ABC = 1/2 | (x₁ y₂ + x₂ y₃ + x₃ y₁) - (x₂ y₁ + x₃ y₂ + x₁ y₃) | sq. units.

1. Find the area of the triangle whose vertices have co-ordinates:

(i) (3, 2), (5, 4), (2, 2)

(ii) (6, 2), (- 3, 4), (4, - 3)

(iii) (0, 0), (a cos α, a sin α), (a cos β, a sin β)

(iv) (a cos α, b sin α), (a cos β, a sin β) , (a cos γ, b sin γ)

(v) (at₁², 2at₁), (at₂², 2at₂), (at₃², 2at₃)

(vi) (ct₁, c/t₁), (ct₂, c/t₂), (ct₃, c/t₃).

2. The area of the triangle formed by joining the points (2, 7), (5, 1) and (x, 3) is 18 sq. units. Find x.

3. The polar co-ordinates of the vertices of a triangle are (1, 5π/6), (2, π/2) and (3, π/6); find the area of the triangle.

4. If the polar co-ordinates of the points A, B ,C, D be (2√2, π/4), (4/√3, 2π/3) and (2√2, -5π/4) respectively, then show that the points A, B, C are collinear.

Answers for the worksheet on co-ordinate triangle are given below to check the exact answers of the above questions for finding the area of a triangle.

(i) 1 sq. units

(ii) 24.5 sq. units

(iii) a²/2 |sin⁡(α - β)| sq units

(iv) 2 ab |sin⁡ (α - β)/2 sin⁡ (β - γ)/2 sin (γ - α)/2| sq units

(v) a² |(t₁ - t₂)(t₂ - t₃)(t₃ - t₁)| sq units

2. 10 or (- 2)

3. 5√3/4 sq. units.

Co-ordinate Geometry

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