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The line segment joining the points (3, -4) and (1, 2) is trisected at the points P and Q . If the coordinates of P and Q are (p, -2) and (\(\frac{5}3\), q) respectively. Find the values of p and q.

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Let given points be A(3, -4) and B(1, 2) , which is trisected at points P(p, -2) and Q(\(\frac{5}3\), q).

By section formula,

x = \(\frac{mx_2+nx_1}{m+n}\), y = \(\frac{my_2+ny_1}{m+n}\)

As point P divides the line in 1:2 and Q divides the line in 2:1. 

For point P(p, -2)of AB, where m = 1 and n = 2,

p = \(\frac{1\times1+3\times2}{1+2}\), -2 = \(\frac{2\times1+2\times(-4)}{1+2}\)

Solving for p,

p = \(\frac{7}3\)

For point Q(5/3, q) of AB, where m = 2 and n = 1,

\(\frac{5}3\) = \(\frac{2\times1+1\times3}{2+1}\),q = \(\frac{2\times2+1\times(-4)}{2+1}\)

Solving for q,

q = \(\frac{4-4}3\)

∴ q = 0Hence, the value of p and q are \(\frac{7}3\) and 0 respectively.

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