In the worksheet on coordinate triangle we need to find the area of a triangle where the three coordinates 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 coordinates 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 coordinates (x₁, y₁), (x₂, y₂) and (x₃, y₃) of the vertices A, B, C respectively.
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∆ ABC = 1/2  (x₁ y₂ + x₂ y₃ + x₃ y₁)  (x₂ y₁ + x₃ y₂ + x₁ y₃)  sq. units.
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1. Find the area of the triangle whose vertices have coordinates:
(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 coordinates 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 coordinates 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 coordinate 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.
● Coordinate Geometry
11 and 12 Grade Math
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