find the point which is three-fourth of the way from (3,1) to (-2,5)

Let point P is three - fourth on the way from A (3, 1) to B (– 2, 5)

$\Rightarrow \frac{AP}{AB} = \frac{3}{4} \Rightarrow \frac{AB}{AP} = \frac{4}{3} \Rightarrow \frac{AB}{AP} - 1 = \frac{4}{3} - 1 \Rightarrow \frac{AB - AP}{AP} = \frac{4 - 3}{3} \Rightarrow \frac{BP}{AP} = \frac{1}{3} \Rightarrow \frac{AP}{BP} = \frac{2}{1} \Rightarrow AP : BP = 3 : 1$

Hence coordinates of P are

$(\frac{3 \times - 2 + 1 \times 3}{3 + 1}, \frac{3 \times 5 + 1 \times 1}{3 + 1}) = (\frac{- 6 + 3}{4}, \frac{15 + 1}{4}) = (- \frac{3}{4}, 4)$

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