To solve the given statements, we will analyze each statement one by one and determine if they are true (T) or false (F).
### Statement P:
The point \(\left(\frac{7}{4}, \frac{7}{8}\right)\) divides the line segment joining the points (4, -1) and (-2, 4) internally in the ratio 3:5.
**Step 1: Use the section formula to find the coordinates of the point dividing the line segment.**
The section formula states that if a point \(P\) divides the line segment joining points \(A(x_1, y_1)\) and \(B(x_2, y_2)\) in the ratio \(m:n\), then the coordinates of point \(P\) are given by:
\[
P\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right)
\]
Here, \(A(4, -1)\), \(B(-2, 4)\), \(m = 3\), and \(n = 5\).
**Step 2: Calculate the x-coordinate.**
\[
x = \frac{3 \cdot (-2) + 5 \cdot 4}{3 + 5} = \frac{-6 + 20}{8} = \frac{14}{8} = \frac{7}{4}
\]
**Step 3: Calculate the y-coordinate.**
\[
y = \frac{3 \cdot 4 + 5 \cdot (-1)}{3 + 5} = \frac{12 - 5}{8} = \frac{7}{8}
\]
**Conclusion for Statement P:**
The coordinates of the point that divides the segment are \(\left(\frac{7}{4}, \frac{7}{8}\right)\), which matches the given point. Therefore, Statement P is **True (T)**.
### Statement Q:
The point (5, 0) on the y-axis is equidistant from (-1, 2) and (3, 4).
**Step 1: Check the coordinates of point (5, 0).**
The point (5, 0) is not on the y-axis; it is on the x-axis since its x-coordinate is 5.
**Conclusion for Statement Q:**
Since (5, 0) is not on the y-axis, Statement Q is **False (F)**.
### Statement R:
The points (8, 1), (3, -4), and (2, -5) are collinear.
**Step 1: Use the collinearity condition.**
Three points are collinear if the area of the triangle formed by them is zero. We can use the determinant method:
\[
\text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|
\]
Substituting the points (8, 1), (3, -4), and (2, -5):
\[
\text{Area} = \frac{1}{2} \left| 8(-4 + 5) + 3(-5 - 1) + 2(1 + 4) \right|
\]
Calculating:
\[
= \frac{1}{2} \left| 8(1) + 3(-6) + 2(5) \right| = \frac{1}{2} \left| 8 - 18 + 10 \right| = \frac{1}{2} \left| 0 \right| = 0
\]
**Conclusion for Statement R:**
Since the area is zero, the points are collinear. Therefore, Statement R is **True (T)**.
### Statement S:
The centroid of the triangle whose vertices are (1, 4), (-1, -1), and (3, -2) is \((1, \frac{1}{3})\).
**Step 1: Calculate the centroid.**
The centroid \(G\) of a triangle with vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) is given by:
\[
G\left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right)
\]
Substituting the vertices (1, 4), (-1, -1), and (3, -2):
\[
G\left(\frac{1 + (-1) + 3}{3}, \frac{4 + (-1) + (-2)}{3}\right) = G\left(\frac{3}{3}, \frac{1}{3}\right) = G(1, \frac{1}{3})
\]
**Conclusion for Statement S:**
The calculated centroid matches the given point. Therefore, Statement S is **True (T)**.
### Final Results:
- Statement P: T
- Statement Q: F
- Statement R: T
- Statement S: T
### Summary:
The final answers are:
- P: T
- Q: F
- R: T
- S: T
