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# State whether the following statements are true or false. Justify your answer.

Point $ P(5,-3) $ is one of the two points of trisection of the line segment joining the points $ A(7,-2) $ and $ B(1,-5) $.

Given:

Point \( P(5,-3) \) is one of the two points of trisection of the line segment joining the points \( A(7,-2) \) and \( B(1,-5) \).

To do:

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

Solution:

Let $\mathrm{P}(5,-3)$ divides the line segment joining the points $A(7,-2)$ and $B(1,-5)$ in the ratio k: 1 internally.

Using section formula, we get,

$P(5,-3)=(\frac{k(1)+(1)(7)}{k+1}, \frac{k(-5)+1(-2)}{k+1})$

$=(\frac{k+7}{k+1}, \frac{-5 k-2}{k+1})$

This implies,

$5=\frac{k+7}{k+1}$ and $-3=\frac{-5 k-2}{k+1}$

$5(k+1)=k+7$

$5 k+5 =k+7$

$5k-k=7-5$

$4k=2$

$k=\frac{2}{4}$

$k=\frac{1}{2}$

Therefore, the point P divides the line segment AB in ratio 1: 2.

Hence, point P is the point of trisection of AB.

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