The point \(P(5,-3)\) is one of the two trisection points of the segment joining \(A(7,-2)\) and \(B(1,-5)\).
True.
Step 1: Recall what trisection means.
Trisection of a line segment means dividing it into 3 equal parts. So, two points divide the line segment in the ratio \(1:2\) and \(2:1\).
Step 2: Formula for internal division of a line.
If a point \(P(x, y)\) divides the line joining two points \((x_1, y_1)\) and \((x_2, y_2)\) in the ratio \(m:n\), then
\[ P(x, y) = \Bigg( \dfrac{m x_2 + n x_1}{m+n},\, \dfrac{m y_2 + n y_1}{m+n} \Bigg) \]
Step 3: Apply the formula for ratio \(1:2\).
Here, we take \(A(7, -2) = (x_1, y_1)\) and \(B(1, -5) = (x_2, y_2)\).
Ratio = \(1:2\). That means \(m = 1, n = 2\).
Step 4: Calculate the x-coordinate.
\[ x = \dfrac{1 \cdot 7 + 2 \cdot 1}{1+2} = \dfrac{7 + 2}{3} = \dfrac{9}{3} = 3 \]
Wait! This is not \(5\). Let’s try ratio \(2:1\) instead (because we need both trisection points).
Step 5: Calculate again for ratio \(2:1\).
Now \(m = 2, n = 1\).
\[ x = \dfrac{2 \cdot 7 + 1 \cdot 1}{2+1} = \dfrac{14 + 1}{3} = \dfrac{15}{3} = 5 \]
Step 6: Calculate the y-coordinate for ratio \(2:1\).
\[ y = \dfrac{2 \cdot (-2) + 1 \cdot (-5)}{2+1} = \dfrac{-4 - 5}{3} = \dfrac{-9}{3} = -3 \]
Step 7: Final Result.
So, the point is \((5, -3)\), which matches the given point \(P(5, -3)\).
Conclusion: Yes, the given point is one of the trisection points. Hence, the statement is True.