Chapter 1: Orienting Yourself: The Use of Coordinates
Chapter 1: Orienting Yourself: The Use of Coordinates
Exercise
Question
Question: 13
20 / 25
20 / 25
The midpoints of the sides of triangle ABC are the points D, E and F. Given that the coordinates of D, E, and F are (5, 1), (6, 5) and (0, 3) respectively, find the coordinates of A, B and C.
SponsoredDesktop Banner
Answer
1
Triangle ABC with D, E, and F marking the midpoints of sides AB, BC, and CA.
Since D is the midpoint of AB, use the midpoint formula for A and B.
\text{$\left(\dfrac{x1 + x2}{2}, \dfrac{y1 + y2}{2}\right) = (5, 1)$}
\text{⇒ (x1 + x2, y1 + y2) = (10, 2)$\quad$ ${color:#625b5f}[Eq 1 · Result after multiplying both sides by 2]{/color}$}
2
Since E is the midpoint of BC, use the midpoint formula for B and C.
\text{$\left(\dfrac{x2 + x3}{2}, \dfrac{y2 + y3}{2}\right) = (6, 5)$}
\text{⇒ (x2 + x3, y2 + y3) = (12, 10) $\quad$ ${color:#625b5f}[Eq 2 · Result after multiplying both sides by 2]{/color}$}
3
Since F is the midpoint of CA, use the midpoint formula for C and A.
\text{$\left(\dfrac{x3 + x1}{2}, \dfrac{y3 + y1}{2}\right) = (0, 3)$}
\text{⇒ (x3 + x1, y3 + y1) = (0, 6) $\quad$ ${color:#625b5f}[Eq 3 · Result after multiplying both sides by 2]{/color}$}
4
Add Eq 1 and Eq 2, then subtract Eq 3 to find the coordinates of B.
((x1 + x2) + (x2 + x3) - (x3 + x1), (y1 + y2) + (y2 + y3) - (y3 + y1)) = (10 + 12 - 0, 2 + 10 - 6)
⇒ (2 × x2, 2 × y2) = (22, 6)
\text{⇒ (x2, y2) = (11, 3)$\quad$ ${color:#625b5f}[Result after dividing both sides by 2]{/color}$}
5
Add Eq 2 and Eq 3, then subtract Eq 1 to find the coordinates of C.
((x2 + x3) + (x3 + x1) - (x1 + x2), (y2 + y3) + (y3 + y1) - (y1 + y2)) = (12 + 0 - 10, 10 + 6 - 2)
⇒ (2 × x3, 2 × y3) = (2, 14)
\text{⇒ (x3, y3) = (1, 7)$\quad$ ${color:#625b5f}[Result after dividing both sides by 2]{/color}$}
6
Add Eq 3 and Eq 1, then subtract Eq 2 to find the coordinates of A.
((x3 + x1) + (x1 + x2) - (x2 + x3), (y3 + y1) + (y1 + y2) - (y2 + y3)) = (0 + 10 - 12, 6 + 2 - 10)
⇒ (2 × x1, 2 × y1) = (-2, -2)
\text{⇒ (x1, y1) = (-1, -1)$\quad$ ${color:#625b5f}[Result after dividing both sides by 2]{/color}$}
7
Use the coordinates found in the previous steps to write the coordinates of A, B, and C.
Answer: A = (-1, -1), B = (11, 3), and C = (1, 7)