What are the coordinates of point P on the directed line segment from $R$ to $Q$ such that $P$ is $\frac{5}{6}$ the length of the line segment from $R$ to $Q$ ? Round to the nearest tenth, if necessary.
Answer (1)
We have points R ( 2 , 1 ) and Q ( 14 , 9 ) , and we want to find point P that is 6 5 of the way from R to Q .
We use the section formula to find the coordinates of P : x P = 6 5 x Q + x R and y P = 6 5 y Q + y R .
Substituting the coordinates of R and Q , we get x P = 6 5 ( 14 ) + 2 = 12 and y P = 6 5 ( 9 ) + 1 = 6 46 ≈ 7.6667 .
Rounding to the nearest tenth, we find the coordinates of P to be ( 12 , 7.7 ) .
Explanation
Analyze the problem We are given two points, R ( 2 , 1 ) and Q ( 14 , 9 ) , and we want to find the coordinates of point P on the directed line segment from R to Q such that P is 6 5 the length of the line segment from R to Q . This means that the point P divides the segment RQ in the ratio 5 : 1 .
Apply the section formula Let the coordinates of point P be ( x P , y P ) . We can use the section formula to find the coordinates of P . The section formula states that if a point P divides the line segment joining points R ( x R , y R ) and Q ( x Q , y Q ) in the ratio m : n , then the coordinates of P are given by:
x P = m + n m x Q + n x R y P = m + n m y Q + n y R
In our case, m = 5 and n = 1 , so the coordinates of P are:
x P = 5 + 1 5 x Q + 1 x R = 6 5 x Q + x R y P = 5 + 1 5 y Q + 1 y R = 6 5 y Q + y R
Calculate the coordinates of P Now, we substitute the coordinates of R ( 2 , 1 ) and Q ( 14 , 9 ) into the formulas for x P and y P :
x P = 6 5 ( 14 ) + 2 = 6 70 + 2 = 6 72 = 12 y P = 6 5 ( 9 ) + 1 = 6 45 + 1 = 6 46 = 3 23 ≈ 7.6667
So, the coordinates of point P are ( 12 , 3 23 ) .
Round to the nearest tenth We are asked to round the coordinates to the nearest tenth, if necessary. Since x P = 12 is already an integer, we don't need to round it. For y P , we have y P = 3 23 ≈ 7.6667 . Rounding to the nearest tenth, we get y P ≈ 7.7 .
Therefore, the coordinates of point P are approximately ( 12 , 7.7 ) .
State the final answer The coordinates of point P on the directed line segment from R to Q such that P is 6 5 the length of the line segment from R to Q are approximately ( 12 , 7.7 ) .
Examples
In computer graphics, when drawing a line from one point to another, you might want to find a point that is a certain fraction of the way along that line. This is useful for creating smooth animations or placing objects at specific locations along a path. For example, if you have a starting point and an ending point, you can use the section formula to calculate the coordinates of a point that is, say, 5/6 of the way from the start to the end, allowing you to precisely position an element in your animation or design.
