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.

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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.

Answers:

(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
  • What is Co-ordinate Geometry?
  • Rectangular Cartesian Co-ordinates
  • Polar Co-ordinates
  • Relation between Cartesian and Polar Co-Ordinates
  • Distance between Two given Points
  • Distance between Two Points in Polar Co-ordinates
  • Division of Line Segment: Internal & External
  • Area of the Triangle Formed by Three co-ordinate Points
  • Condition of Collinearity of Three Points
  • Medians of a Triangle are Concurrent
  • Apollonius' Theorem
  • Quadrilateral form a Parallelogram
  • Problems on Distance Between Two Points
  • Area of a Triangle Given 3 Points
  • Worksheet on Quadrants
  • Worksheet on Rectangular – Polar Conversion
  • Worksheet on Line-Segment Joining the Points
  • Worksheet on Distance Between Two Points
  • Worksheet on Distance Between the Polar Co-ordinates
  • Worksheet on Finding Mid-Point
  • Worksheet on Division of Line-Segment
  • Worksheet on Centroid of a Triangle
  • Worksheet on Area of Co-ordinate Triangle
  • Worksheet on Collinear Triangle
  • Worksheet on Area of Polygon
  • Worksheet on Cartesian Triangle

  • 11 and 12 Grade Math

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