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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?​

1 Answer

6 votes

Answer:

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

User Yash Tewari
by
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