The problem asks us to find the midpoint of the segment PQ, where P and Q are points on the line segment joining A(-2, 5) and B(3, 1) such that AP = PQ = QB.

Understanding the Division of the Line Segment AB

Since AP = PQ = QB, the points P and Q divide the line segment AB into three equal parts. This means P and Q are the points of trisection of AB.

• P is the point that divides AB in the ratio AP : PB. Since AP = PQ = QB, let the length of each part be \(k\). Then AP = \(k\), PQ = \(k\), and QB = \(k\). The length of PB is PQ + QB = \(k + k = 2k\). Thus, P divides AB in the ratio \(k : 2k = 1 : 2\).
• Q is the point that divides AB in the ratio AQ : QB. The length of AQ is AP + PQ = \(k + k = 2k\). The length of QB is \(k\). Thus, Q divides AB in the ratio \(2k : k = 2 : 1\).

Calculating the Coordinates of Point P

We use the section formula to find the coordinates of P(\(x_P, y_P\)) which divides the line segment joining A(\(x_1, y_1\)) = (-2, 5) and B(\(x_2, y_2\)) = (3, 1) in the ratio \(m:n = 1:2\).

The section formula is given by:

\((x, y) = \left( \frac{m x_2 + n x_1}{m+n}, \frac{m y_2 + n y_1}{m+n} \right)\)

For point P, \(m=1\) and \(n=2\):

\(x_P = \frac{1 \times 3 + 2 \times (-2)}{1 + 2} = \frac{3 - 4}{3} = \frac{-1}{3}\)

\(y_P = \frac{1 \times 1 + 2 \times 5}{1 + 2} = \frac{1 + 10}{3} = \frac{11}{3}\)

So, the coordinates of P are \(\left(-\frac{1}{3}, \frac{11}{3}\right)\).

Calculating the Coordinates of Point Q

We use the section formula to find the coordinates of Q(\(x_Q, y_Q\)) which divides the line segment joining A(\(x_1, y_1\)) = (-2, 5) and B(\(x_2, y_2\)) = (3, 1) in the ratio \(m:n = 2:1\).

For point Q, \(m=2\) and \(n=1\):

\(x_Q = \frac{2 \times 3 + 1 \times (-2)}{2 + 1} = \frac{6 - 2}{3} = \frac{4}{3}\)

\(y_Q = \frac{2 \times 1 + 1 \times 5}{2 + 1} = \frac{2 + 5}{3} = \frac{7}{3}\)

So, the coordinates of Q are \(\left(\frac{4}{3}, \frac{7}{3}\right)\).

Calculating the Midpoint of Segment PQ

Now we need to find the midpoint of the line segment PQ. Let the midpoint be M(\(x_M, y_M\)). We use the midpoint formula:

\((x_M, y_M) = \left( \frac{x_P + x_Q}{2}, \frac{y_P + y_Q}{2} \right)\)

Substitute the coordinates of P \(\left(-\frac{1}{3}, \frac{11}{3}\right)\) and Q \(\left(\frac{4}{3}, \frac{7}{3}\right)\):

\(x_M = \frac{-\frac{1}{3} + \frac{4}{3}}{2} = \frac{\frac{-1 + 4}{3}}{2} = \frac{\frac{3}{3}}{2} = \frac{1}{2}\)

\(y_M = \frac{\frac{11}{3} + \frac{7}{3}}{2} = \frac{\frac{11 + 7}{3}}{2} = \frac{\frac{18}{3}}{2} = \frac{6}{2} = 3\)

So, the midpoint of PQ is \(\left(\frac{1}{2}, 3\right)\).

Summary of Steps

• Identify that P and Q are points of trisection dividing AB into three equal parts.
• Determine the ratio in which P and Q divide AB. P divides in 1:2, Q divides in 2:1.
• Use the section formula to find the coordinates of P.
• Use the section formula to find the coordinates of Q.
• Use the midpoint formula to find the midpoint of the segment PQ.

The coordinates of the midpoint of PQ are \(\left(\frac{1}{2}, 3\right)\).