Question

Co-ordinate Geometry

for what value of a the area of triangle formed by the points (2, 4), (6. a) and
|-1, 1) is 9 sq. units?​

Answer 1

The value of a = 14

Explanation:

Given

(x₁, y₁) = (2, 4)

(x₂, y₂) = (6, a)

(x₃, y₃) = (-1, 1)

A = 9 sq.units

Area of a triangle with vertices (x₁, y₁), (x₂, y₂), and (x₃, y₃) is:

`A=(\left|x_1\left(y_2-y_3\right)+x_2\left(y_3-y_1\right)+x_3\left(y_1-y_2\right)\right|)/(2)`

substituting the values (x₁, y₁) = (2, 4), (x₂, y₂) = (6, a), (x₃, y₃) = (-1, 1), A = 9 in th formula

`A=(\left|x_1\left(y_2-y_3\right)+x_2\left(y_3-y_1\right)+x_3\left(y_1-y_2\right)\right|)/(2)`

`9=(\left|2\left(a-1\right)+6\left(1-4\right)+-1\left(4-a\right)\right|)/(2)`

Multiply both sides by 2

`(2\left|2\left(a-1\right)+6\left(1-4\right)-1\left(4-a\right)\right|)/(2)=9\cdot \:2`

simplify

`\left|2\left(a-1\right)+6\left(1-4\right)-1\left(4-a\right)\right|=18`

As the area is always positive.

so

`2\left(a-1\right)+6\left(1-4\right)-1\cdot \left(4-a\right)=18`

`2\left(a-1\right)-18-\left(4-a\right)=18`

Add 18 to both sides

`2\left(a-1\right)-18-\left(4-a\right)+18=18+18`

simplify

`2\left(a-1\right)-\left(4-a\right)=36`

`3a-6=36`

`3a=42`

Divide both sides by 3

`(3a)/(3)=(42)/(3)`

`a=14`

Thus, the value of a = 14