Orienting Yourself: The Use of Coordinates | EOT

Question 11

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 $B(16, -2)$ .

Solution

Understand the Question

Points of trisection $P$ and $Q$ divide the line segment $AB$ into three equal parts such that $AP = PQ = QB$ .
Since $AP = PQ$ , $P$ is the midpoint of the segment $AQ$ .
Since $PQ = QB$ , $Q$ is the midpoint of the segment $PB$ .
Using the midpoint formula $\left(\dfrac{x_1 + x_2}{2}, \dfrac{y_1 + y_2}{2}\right)$ , we can set up and solve a system of linear equations to find the coordinates of $P$ and $Q$ for $A(4, 7)$ and $B(16, -2)$ .

Step 1 · Set Up Midpoint Equations for Coordinates

Let the coordinates of $P$ be $(x_1, y_1)$ and $Q$ be $(x_2, y_2)$ .Since $P$ is the midpoint of $AQ$ :

x_1 = \dfrac{4 + x_2}{2} \implies 2x_1 - x_2 = 4 \quad \dots (1)

y_1 = \dfrac{7 + y_2}{2} \implies 2y_1 - y_2 = 7 \quad \dots (2)

Since $Q$ is the midpoint of $PB$ :

x_2 = \dfrac{x_1 + 16}{2} \implies 2x_2 - x_1 = 16 \quad \dots (3)

y_2 = \dfrac{y_1 + (-2)}{2} \implies 2y_2 - y_1 = -2 \quad \dots (4)

Step 2 · Solve for the Coordinates of $P$ and $Q$

From equation $(1)$ , $x_2 = 2x_1 - 4$ . Substitute this into equation $(3)$ :

\begin{aligned} 2(2x_1 - 4) - x_1 &= 16 \\ 4x_1 - 8 - x_1 &= 16 \\ 3x_1 &= 24 \\ x_1 &= 8 \end{aligned}

Substitute $x_1 = 8$ back into equation $(1)$ :

x_2 = 2(8) - 4 = 12

From equation $(2)$ , $y_2 = 2y_1 - 7$ . Substitute this into equation $(4)$ :

\begin{aligned} 2(2y_1 - 7) - y_1 &= -2 \\ 4y_1 - 14 - y_1 &= -2 \\ 3y_1 &= 12 \\ y_1 &= 4 \end{aligned}

Substitute $y_1 = 4$ back into equation $(2)$ :

y_2 = 2(4) - 7 = 1

Therefore, the coordinates are $P(8, 4)$ and $Q(12, 1)$ .

Answer

$P(8, 4)$ and $Q(12, 1)$

Common Mistakes

Ignoring the Midpoint Method Constraint: Using the section formula with ratios $1:2$ and $2:1$ directly instead of using the midpoint formula as specifically required by the question.
Incorrect Midpoint Relationships: Assuming $P$ and $Q$ are midpoints of $AB$ , rather than recognizing $P$ is the midpoint of $AQ$ and $Q$ is the midpoint of $PB$ .
Sign Errors in Linear Equations: Making algebraic errors when rearranging and substituting expressions containing negative coordinates (such as the $y$ -coordinate of $B$ , which is $-2$ ).

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

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?