The following table shows the coordinates of points , and . In each case, state whether is the midpoint of segment . Justify your answer. When is the mid-point of , can you find any connection between the coordinates of , and ?

Understand the Question A point is the midpoint of a segment if it lies on the line segment connecting and and divides it into two equal parts (). In coordinate geometry, the midpoint represents the exact center between two points. The coordinates of the midpoint are the averages (arithmetic means) of the corresponding coordinates of the endpoints. Step 1 · Condition for a Point to be a Midpoint For a point to be the midpoint of a line segment joining and :1. Points , , and must be collinear with lying between and . 2. The distances must be equal: Using the distance formula: Step 2 · Derive the Connection Between Coordinates When is the midpoint of segment : The -coordinate of is the average of the -coordinates of and : The -coordinate of is the average of the -coordinates of and : Hence, the coordinates of midpoint are: Answer > is the midpoint of if and only if its coordinates are the averages of the coordinates of and : Common Mistakes Subtracting instead of adding: Confusing the midpoint formula with the distance formula and writing instead of . Forgetting to divide by 2: Adding the coordinates without dividing by . Mixing and coordinates: Combining the -coordinate of one point with the -coordinate of another.

Orienting Yourself: The Use of Coordinates | EOT

Question 9

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 $T$ ?

Solution

Understand the Question

A point $M$ is the midpoint of a segment $ST$ if it lies on the line segment connecting $S$ and $T$ and divides it into two equal parts ( $SM = MT$ ).
In coordinate geometry, the midpoint represents the exact center between two points.
The coordinates of the midpoint are the averages (arithmetic means) of the corresponding coordinates of the endpoints.

Step 1 · Condition for a Point to be a Midpoint

For a point $M(x, y)$ to be the midpoint of a line segment joining $S(x_1, y_1)$ and $T(x_2, y_2)$ :1. Points $S$ , $M$ , and $T$ must be collinear with $M$ lying between $S$ and $T$ . 2. The distances must be equal: $SM = MT = \dfrac{1}{2}ST$

Using the distance formula:

SM = \sqrt{(x - x_1)^2 + (y - y_1)^2}

MT = \sqrt{(x_2 - x)^2 + (y_2 - y)^2}

Step 2 · Derive the Connection Between Coordinates

When $M(x, y)$ is the midpoint of segment $ST$ :

The $x$ -coordinate of $M$ is the average of the $x$ -coordinates of $S$ and $T$ :

x = \dfrac{x_1 + x_2}{2}

The $y$ -coordinate of $M$ is the average of the $y$ -coordinates of $S$ and $T$ :

y = \dfrac{y_1 + y_2}{2}

Hence, the coordinates of midpoint $M$ are:

M = \left(\dfrac{x_1 + x_2}{2}, \dfrac{y_1 + y_2}{2}\right)

Answer

$M$ is the midpoint of $ST$ if and only if its coordinates are the averages of the coordinates of $S$ and $T$ : $M = \left(\dfrac{x_S + x_T}{2}, \dfrac{y_S + y_T}{2}\right)$

Common Mistakes

Subtracting instead of adding: Confusing the midpoint formula with the distance formula and writing $\dfrac{x_2 - x_1}{2}$ instead of $\dfrac{x_1 + x_2}{2}$ .
Forgetting to divide by 2: Adding the coordinates $(x_1 + x_2, y_1 + y_2)$ without dividing by $2$ .
Mixing $x$ and $y$ coordinates: Combining the $x$ -coordinate of one point with the $y$ -coordinate of another.

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

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 $C$ .

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?