Question

fractional distance
given the points j(4,-7) and l(-2,13), find the coordinates of point k on jl such that the ratio of jk to jl is 1:4.
given the points r(6,-2) and t(-9,-7), find the coordinates of point s on rt such that the ratio of rs to st is 3:2.

Explanation:

1. For the first - part (finding the coordinates of point \(K\) on \(JL\) with \(JK:JL = 1:4\)):

• The section - formula for finding the coordinates of a point \(P(x,y)\) that divides the line - segment joining \(A(x_1,y_1)\) and \(B(x_2,y_2)\) in the ratio \(m:n\) is \(x=\frac{mx_2+nx_1}{m + n}\) and \(y=\frac{my_2+ny_1}{m + n}\). Here, since \(JK:JL = 1:4\), then \(JK:KL=1:3\) (because \(JL=JK + KL\)). So, \(m = 1\) and \(n = 3\), \(x_1 = 4,y_1=-7,x_2=-2,y_2 = 13\).
• Step 1: Calculate the \(x\) - coordinate of \(K\)
• Using the formula \(x=\frac{mx_2+nx_1}{m + n}\), substitute \(m = 1\), \(n = 3\), \(x_1 = 4\), and \(x_2=-2\).
• \(x=\frac{1\times(-2)+3\times4}{1 + 3}=\frac{-2 + 12}{4}=\frac{10}{4}=\frac{5}{2}\).
• Step 2: Calculate the \(y\) - coordinate of \(K\)
• Using the formula \(y=\frac{my_2+ny_1}{m + n}\), substitute \(m = 1\), \(n = 3\), \(y_1=-7\), and \(y_2 = 13\).
• \(y=\frac{1\times13+3\times(-7)}{1 + 3}=\frac{13-21}{4}=\frac{-8}{4}=-2\).
• So, the coordinates of \(K\) are \((\frac{5}{2},-2)\).

1. For the second - part (finding the coordinates of point \(S\) on \(RT\) with \(RS:ST = 3:2\)):

• Here, \(m = 3\), \(n = 2\), \(x_1 = 6,y_1=-2,x_2=-9,y_2=-7\).
• Step 1: Calculate the \(x\) - coordinate of \(S\)
• Using the formula \(x=\frac{mx_2+nx_1}{m + n}\), substitute \(m = 3\), \(n = 2\), \(x_1 = 6\), and \(x_2=-9\).
• \(x=\frac{3\times(-9)+2\times6}{3 + 2}=\frac{-27 + 12}{5}=\frac{-15}{5}=-3\).
• Step 2: Calculate the \(y\) - coordinate of \(S\)
• Using the formula \(y=\frac{my_2+ny_1}{m + n}\), substitute \(m = 3\), \(n = 2\), \(y_1=-2\), and \(y_2=-7\).
• \(y=\frac{3\times(-7)+2\times(-2)}{3 + 2}=\frac{-21-4}{5}=\frac{-25}{5}=-5\).
• So, the coordinates of \(S\) are \((-3,-5)\).

Answer:

The coordinates of \(K\) are \((\frac{5}{2},-2)\) and the coordinates of \(S\) are \((-3,-5)\)