Chapter 1: Orienting Yourself: The Use of Coordinates

Exercise

Question

Question: 11

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Let P, Q be points of trisection of AB, with P closer to A, and Q closer to B. Using your knowledge of how to find the coordinates of the midpoint of a segment, how would you find coordinates of P and Q? Do this for the case when the points are A (4, 7) and B (16, -2).

Answer

1
Think of AB as being divided into three equal parts. Since P is closer to A and Q is closer to B, P is one-third of the way from A to B, while Q is two-thirds of the way from A to B.
AP = PQ = QB
Line segment AB divided into three equal parts by points P and Q.
Line segment AB divided into three equal parts by P and Q.
2
Find how much the x- and y-coordinates change as we move from A to B.
\text{$\Delta$x = 16 - 4 = 12 $\quad$ ${color:#625b5f}[x1 = 4, x2 = 16]{/color}$}
\text{Δy = (-2) - 7 = -9 $\quad$ ${color:#625b5f}[y1 = 7, y2 = -2]{/color}$}
3
Since P is one-third of the way from A to B, find one-third of the change in the x- and y-coordinates from A to B and add these to the coordinates of A.
\text{x_P = 4 + $\dfrac{\Delta x}{3} = 4 + \dfrac{12}{3} = 4 + 4 = 8$ $\quad$ ${color:#625b5f}[x(A) = 4]{/color}$}
\text{y_P = 7 + $\dfrac{\Delta y}{3} = 7 + \dfrac{-9}{3} = 7 + (-3) = 4$ $\quad$ ${color:#625b5f}[y(A) = 7]{/color}$}
4
Since Q is two-thirds of the way from A to B, find two-thirds of the change in the x- and y-coordinates from A to B and add these to the coordinates of A.
\text{x_Q = 4 + $\dfrac{2 × \Delta x}{3} = 4 + \dfrac{2 × (12)}{3} = 4 + 8 = 12$ $\quad$ ${color:#625b5f}[x(A) = 4]{/color}$}
\text{y_Q = 7 + $\dfrac{2 × \Delta y}{3} = 7 + \dfrac{2 × (-9)}{3} = 7 + (-6) = 1$ $\quad$ ${color:#625b5f}[y(A) = 7]{/color}$}
5
Use the coordinates found for P and Q to write the final answer.
Answer: P = (8, 4), Q = (12, 1)
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