Orienting Yourself: The Use of Coordinates | EOT

Question 8

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.

Solution

Understand the Question

The origin $O(0, 0)$ is fixed as one of the three vertices of each triangle.
An isosceles triangle has at least two sides of equal length.
In a right-angled isosceles triangle, the two perpendicular sides meeting at the $90^\circ$ vertex are equal in length.
For signs of coordinates across quadrants:
- Quadrant III: $x < 0$ and $y < 0 \implies (-a, -b)$
- Quadrant IV: $x > 0$ and $y < 0 \implies (a, -b)$

(i) A right-angled isosceles triangle.

Step 1 · Choose Coordinates for the Vertices

Let one vertex be at the origin $O(0, 0)$ .To form a right-angled isosceles triangle with the right angle at the origin:

Place the second vertex along the positive $x$ -axis: $A(a, 0)$
Place the third vertex along the positive $y$ -axis: $B(0, a)$

The lengths of the perpendicular sides are:

\begin{aligned} OA &= a \\[0.6em] OB &= a \end{aligned}

Since $OA = OB = a$ and $\angle AOB = 90^\circ$ , $\Delta OAB$ is a right-angled isosceles triangle.

For example, choosing $a = 4$ , the vertices are $O(0, 0)$ , $A(4, 0)$ , and $B(0, 4)$ .

Answer

(i) Vertices: $O(0, 0)$ , $A(a, 0)$ , and $B(0, a)$ (e.g., $(0, 0)$ , $(4, 0)$ , and $(0, 4)$ )

(ii) An isosceles triangle with one vertex in Quadrant III and the other in Quadrant IV.

Step 1 · Choose Coordinates in Quadrants III and IV

Let the first vertex be at the origin $O(0, 0)$ .For the remaining two vertices:

Let vertex $P$ lie in Quadrant III: $P(-a, -b)$ , where $a > 0, b > 0$ .
Let vertex $Q$ lie in Quadrant IV: $Q(a, -b)$ , where $a > 0, b > 0$ .

Using the distance formula from the origin $O(0, 0)$ :

\begin{aligned} OP &= \sqrt{(-a - 0)^2 + (-b - 0)^2} = \sqrt{a^2 + b^2} \\[0.6em] OQ &= \sqrt{(a - 0)^2 + (-b - 0)^2} = \sqrt{a^2 + b^2} \end{aligned}

Since $OP = OQ$ , $\Delta OPQ$ is an isosceles triangle with $P$ in Quadrant III and $Q$ in Quadrant IV.

For example, choosing $a = 3$ and $b = 4$ , the vertices are $O(0, 0)$ , $P(-3, -4)$ , and $Q(3, -4)$ , giving equal sides $OP = OQ = 5$ .

Answer

(ii) Vertices: $O(0, 0)$ , $P(-a, -b)$ , and $Q(a, -b)$ (e.g., $(0, 0)$ , $(-3, -4)$ , and $(3, -4)$ )

Common Mistakes

Quadrant Sign Error: Confusing signs between Quadrants III ( $(-, -)$ ) and IV ( $(+, -)$ ).
Unequal Side Lengths: Selecting arbitrary points in Quadrants III and IV without ensuring $OP = OQ = \sqrt{x^2 + y^2}$ .

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

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?