The midpoints of the sides of triangle are the points , , and . Given that the coordinates of , , and are , , and , respectively, find the coordinates of , , and .

Understand the Question Let the vertices of be , , and . Let , , and be the midpoints of sides , , and respectively. The midpoint of a line segment joining and is given by: Using the midpoint coordinates, we set up systems of linear equations for the -coordinates and -coordinates separately and solve for each vertex. Step 1 · Set Up Equations for the Coordinates Let the vertices of be , , and . Since is the midpoint of : Since is the midpoint of : Since is the midpoint of : Step 2 · Find the x-coordinates Adding equations , , and : Subtracting each equation from : Step 3 · Find the y-coordinates Adding equations , , and : Subtracting each equation from : Answer > Common Mistakes Midpoint Pairing: Inconsistently assigning midpoints to sides. Ensure , , and correspond to the correct opposite vertices (, , and ). Sign Errors in Subtraction: Making arithmetic mistakes when finding negative values like or . Parallelogram Property Shortcut: Since forms a parallelogram, you can verify vertices directly using vertex vector addition: .

Orienting Yourself: The Use of Coordinates | EOT

Question 13

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 $C$ .

Solution

Understand the Question

Let the vertices of $\Delta ABC$ be $A(x_1, y_1)$ , $B(x_2, y_2)$ , and $C(x_3, y_3)$ .
Let $D(5, 1)$ , $E(6, 5)$ , and $F(0, 3)$ be the midpoints of sides $BC$ , $CA$ , and $AB$ respectively.
The midpoint of a line segment joining $(x_a, y_a)$ and $(x_b, y_b)$ is given by: $\left(\dfrac{x_a + x_b}{2}, \dfrac{y_a + y_b}{2}\right)$
Using the midpoint coordinates, we set up systems of linear equations for the $x$ -coordinates and $y$ -coordinates separately and solve for each vertex.

Step 1 · Set Up Equations for the Coordinates

Let the vertices of $\Delta ABC$ be $A(x_1, y_1)$ , $B(x_2, y_2)$ , and $C(x_3, y_3)$ .

Since $D(5, 1)$ is the midpoint of $BC$ : $\dfrac{x_2 + x_3}{2} = 5 \implies x_2 + x_3 = 10 \quad \dots (1)$ $\dfrac{y_2 + y_3}{2} = 1 \implies y_2 + y_3 = 2 \quad \dots (2)$

Since $E(6, 5)$ is the midpoint of $CA$ : $\dfrac{x_3 + x_1}{2} = 6 \implies x_3 + x_1 = 12 \quad \dots (3)$ $\dfrac{y_3 + y_1}{2} = 5 \implies y_3 + y_1 = 10 \quad \dots (4)$

Since $F(0, 3)$ is the midpoint of $AB$ : $\dfrac{x_1 + x_2}{2} = 0 \implies x_1 + x_2 = 0 \quad \dots (5)$ $\dfrac{y_1 + y_2}{2} = 3 \implies y_1 + y_2 = 6 \quad \dots (6)$

Step 2 · Find the x-coordinates

Adding equations $(1)$ , $(3)$ , and $(5)$ :

\begin{aligned} (x_2 + x_3) + (x_3 + x_1) + (x_1 + x_2) &= 10 + 12 + 0 \\ 2(x_1 + x_2 + x_3) &= 22 \\ x_1 + x_2 + x_3 &= 11 \quad \dots (7) \end{aligned}

Subtracting each equation from $(7)$ :

\begin{aligned} x_1 &= (x_1 + x_2 + x_3) - (x_2 + x_3) = 11 - 10 = 1 \\ x_2 &= (x_1 + x_2 + x_3) - (x_3 + x_1) = 11 - 12 = -1 \\ x_3 &= (x_1 + x_2 + x_3) - (x_1 + x_2) = 11 - 0 = 11 \end{aligned}

Step 3 · Find the y-coordinates

Adding equations $(2)$ , $(4)$ , and $(6)$ :

\begin{aligned} (y_2 + y_3) + (y_3 + y_1) + (y_1 + y_2) &= 2 + 10 + 6 \\ 2(y_1 + y_2 + y_3) &= 18 \\ y_1 + y_2 + y_3 &= 9 \quad \dots (8) \end{aligned}

Subtracting each equation from $(8)$ :

\begin{aligned} y_1 &= (y_1 + y_2 + y_3) - (y_2 + y_3) = 9 - 2 = 7 \\ y_2 &= (y_1 + y_2 + y_3) - (y_3 + y_1) = 9 - 10 = -1 \\ y_3 &= (y_1 + y_2 + y_3) - (y_1 + y_2) = 9 - 6 = 3 \end{aligned}

Answer

$A(1, 7), \quad B(-1, -1), \quad C(11, 3)$

Common Mistakes

Midpoint Pairing: Inconsistently assigning midpoints to sides. Ensure $D$ , $E$ , and $F$ correspond to the correct opposite vertices ( $BC$ , $CA$ , and $AB$ ).
Sign Errors in Subtraction: Making arithmetic mistakes when finding negative values like $x_2 = 11 - 12 = -1$ or $y_2 = 9 - 10 = -1$ .
Parallelogram Property Shortcut: Since $AFDE$ forms a parallelogram, you can verify vertices directly using vertex vector addition: $A = E + F - D = (6 + 0 - 5, 5 + 3 - 1) = (1, 7)$ .

More questions in EOT

What are the $x$ -coordinate and $y$ -coordinate of the point of intersection of the two axes?

Point $W$ has $x$ -coordinate equal to $-5$ . Can you predict the coordinates of point $H$ which is on the line through $W$ parallel to the $y$ -axis? Which quadrants can $H$ lie in?

Consider the points $R(3, 0)$ , $A(0, -2)$ , $M(-5, -2)$ , and $P(-5, 2)$ . If they are joined in the same order, predict:

(i) Two sides of $RAMP$ that are perpendicular to each other.

(ii) One side of $RAMP$ that is parallel to one of the axes.

(iii) Two points that are mirror images of each other in one axis. Which axis will this be?

Now plot the points and verify your predictions.

Plot point $Z(5, -6)$ on the Cartesian plane. Construct a right-angled triangle $IZN$ and find the lengths of the three sides.

(Comment: Answers may differ from person to person.)

What would a system of coordinates be like if we did not have negative numbers? Would this system allow us to locate all the points on a 2-D plane?

Are the points $M(-3, -4)$ , $A(0, 0)$ and $G(6, 8)$ on the same straight line? Suggest a method to check this without plotting and joining the points.

Use your method (from Problem 6) to check if the points $R(-5, -1)$ , $B(-2, -5)$ and $C(4, -12)$ are on the same straight line. Now plot both sets of points and check your answers.

Using the origin as one vertex, plot the vertices of:

(i) A right-angled isosceles triangle. (ii) An isosceles triangle with one vertex in Quadrant III and the other in Quadrant IV.

The following table shows the coordinates of points $S$ , $M$ and $T$ . In each case, state whether $M$ is the midpoint of segment $ST$ . Justify your answer.

When $M$ is the mid-point of $ST$ , can you find any connection between the coordinates of $M$ , $S$ and $T$ ?

Q10

Use the connection you found to find the coordinates of $B$ given that $M(-7, 1)$ is the midpoint of $A(3, -4)$ and $B(x, y)$ .

Q11

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 the coordinates of $P$ and $Q$ ? Do this for the case when the points are $A(4, 7)$ and $B(16, -2)$ .

Q12

(i) Given the points $A(1, -8)$ , $B(-4, 7)$ and $C(-7, -4)$ , show that they lie on a circle $K$ whose center is the origin $O(0, 0)$ . What is the radius of circle $K$ ?

(ii) Given the points $D(-5, 6)$ and $E(0, 9)$ , check whether $D$ and $E$ lie within the circle, on the circle, or outside the circle $K$ .

Q13

Q14

A city has two main roads which cross each other at the centre of the city. These two roads are along the North-South (N-S) direction and East-West (E-W) direction. All the other streets of the city run parallel to these roads and are $200\text{ m}$ apart. There are 10 streets in each direction.

(i) Using $1\text{ cm} = 200\text{ m}$ , draw a model of the city in your notebook. Represent the roads/streets by single lines.

(ii) There are street intersections in the model. Each street intersection is formed by two streets — one running in the N-S direction and another in the E-W direction. Each street intersection is referred to in the following manner: If the second street running in the N-S direction and 5th street in the E-W direction meet at some crossing, then we call this street intersection $(2, 5)$ . Using this convention, find: (a) how many street intersections can be referred to as $(4, 3)$ . (b) how many street intersections can be referred to as $(3, 4)$ .

Q15

A computer graphics program displays images on a rectangular screen whose coordinate system has the origin at the bottom-left corner. The screen is 800 pixels wide and 600 pixels high. A circular icon of radius 80 pixels is drawn with its centre at the point $A(100, 150)$ . Another circular icon of radius 100 pixels is drawn with its centre at the point $B(250, 230)$ . Determine:

(i) whether any part of either circle lies outside the screen.

(ii) whether the two circles intersect each other.

Q16

Plot the points $A(2, 1)$ , $B(-1, 2)$ , $C(-2, -1)$ , and $D(1, -2)$ in the coordinate plane. Is $ABCD$ a square? Can you explain why? What is the area of this square?