Question

What are the coordinates of the point that is 3/8 of the way from A(-8,-9) to B(24,-1)

Answer 1

$\text{Let the coordinate of the point be P(x,y)}.\\ \\ \text{this point is 3/8 of the way from }A(-8,-9)\text{ to }B(24,-1).\\ \text{so this point is divide the line joining AB in a ratio }3:5\\ \\ \text{by the division ratio, we know that if a point P divides }A(x_1,y_1)\text{ and }B(x_2,y_2)\\ \text{in a ratio m:n, then we have}$

$P=\left ( \frac{mx_2+nx_1}{m+n}, \ \frac{my_2+ny_1}{m+n} \right )\\ \\ \text{so using this, the coordinates of the required point are:}\\ \\ P=\left ( \frac{3(24)+5(-8)}{3+5}, \ \frac{3(-1)+5(-9)}{3+5} \right )\\ \\ \Rightarrow P=\left ( \frac{32}{8}, \ \frac{-48}{8} \right )\\ \\ \Rightarrow P=(4, -6)$

Hence the coordinate of the point that is 3/8 of the way from A to B is: (4, -6)