## 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 (x_{1}, y_{1}), (x_{2}, y_{2}) and (x_{3}, y_{3}) is

½ | y_{1} (x_{2} - x_{3}) + y_{2} (x_{3} - x_{1}) + y_{3} (x_{1} - x_{2}) | sq. units

or, ½ | x_{1} (y_{2} - y_{3}) + x_{2} (y_{3} - y_{1}) + x_{3} (y_{1} - y_{2}) | sq. units.

In terms of polar co-ordinates (x_{1}, y_{1}), (x_{2}, y_{2}) and (x_{3}, y_{3}) of the vertices A, B, C respectively.

∆ ABC = 1/2 | (x_{1} y_{2} + x_{2} y_{3} + x_{3} y_{1}) - (x_{2} y_{1} + x_{3} y_{2} + x_{1} y_{3}) | 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) (at_{1}^{2}, 2at_{1}), (at_{2}^{2}, 2at_{2}), (at_{3}^{2}, 2at_{3})

(vi) (ct_{1}, c/t_{1}), (ct_{2}, c/t_{2}), (ct_{3}, c/t_{3}).

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) a

^{2}/2 |sin(α - β)| sq units

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

(v) a

^{2} |(t

_{1} - t

_{2})(t

_{2} - t

_{3})(t

_{3} - t

_{1})| 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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