Question

A line segment has endpoints P (1, -1) and Q (-9,-6). What are the coordinates for point R, also on the segment, that will place point R 3/5 of the distance from P to Q?

Answer 1

A line segment has the following endpoints

`P(1,-1)\: and\: Q(-9,-6)`

We are asked to find the coordinates of point R, such that the point R will be 3/5 of the distance PQ.

Let us first find the distance from P to Q.

The distance between two points P and Q is given by

`PQ=\sqrt[]{\mleft({x_2-x_1}\mright)^2+\mleft({y_2-y_1}\mright)^2}`

Let us substitute the given points into the above distance formula

`\begin{gathered} PQ=\sqrt[]{({-9_{}-1_{}})^2+({-6_{}-(-1)_{}})^2} \\ PQ=\sqrt[]{({-9_{}-1_{}})^2+({-6_{}+1_{}})^2} \\ PQ=\sqrt[]{({-10})^2+({-5})^2} \\ PQ=\sqrt[]{100^{}+25^{}} \\ PQ=\sqrt[]{125} \\ PQ=11.18 \end{gathered}`

Now let us find the coordinates of point R such that R is equal to 3/5 of PQ.

So the ratio is m:n = 3:5

`R_x=(nx_1+mx_2)/(n+m),R_y=(ny_1+my_2)/(n+m)`

Let us substitute the given values into the above formula

`\begin{gathered} R_x=(5\cdot1+3\cdot(-9))/(3+5),R_y=(5\cdot(-1)+3\cdot(-6))/(3+5) \\ R_x=(5-27)/(8),R_y=(-5-18)/(8) \\ R_x=(22)/(8),R_y=(-23)/(8) \\ R_x=(11)/(4),R_y=(-23)/(8) \end{gathered}`

Therefore, the coordinates of the point R is

`R((11)/(4),-(23)/(8))`