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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