Question 3

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?

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 $T$?

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

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 $B(16, -2)$.

(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 $K$.

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 $C$.

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 $(3, 4)$.

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.

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?