State whether the following statements are true or false. Justify your answer.
Point $ P(-4,2) $ lies on the line segment joining the points $ A(-4,6) $ and $ B(-4,-6) $.

Given:

Point \( P(-4,2) \) lies on the line segment joining the points \( A(-4,6) \) and \( B(-4,-6) \).

To do:

We have to find whether the given statement is true or false.

Solution:

We know that,

If the point \( P(-4,2) \) lies on the line segment joining the points \( A(-4,6) \) and \( B(-4,-6) \), then the area of triangle ABP is 0.

Area of a triangle $=\frac{1}{2}\left[x_{1}\left(y_{2}-y_{3}\right)+x_{2}\left(y_{3}-y_{1}\right)+x_{3}\left(y_{1}-y_{2}\right)\right]$

Therefore,

Area of triangle ABP $=\frac{1}{2}[-4(-6-2)-4(2-6)-4(6+6)]$

$=\frac{1}{2}[-4(-8)-4(-4)-4(12)]$

$=\frac{1}{2}(32+16-48)$

$=0$

The area of triangle ABP is 0.

Therefore, point \( P(-4,2) \) lies on the line segment joining the points \( A(-4,6) \) and \( B(-4,-6) \).

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Updated on: 10-Oct-2022

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