Plot point on the Cartesian plane. Construct a right-angled triangle and find the lengths of the three sides. (Comment: Answers may differ from person to person.)

Understand the Question To plot , move units right along the positive -axis and units down along the negative -axis (Quadrant IV). To form a right-angled triangle , choose two other points and such that one side is vertical and one side is horizontal, making the angle between them . The lengths of the sides can be determined using coordinate differences and the Pythagoras theorem. Note: Different choices for points and are valid, so side lengths may vary. Step 1 · Plot Point Z and Choose Vertices I and N Plot the point in the fourth quadrant of the Cartesian coordinate system.To construct a right-angled triangle with : Choose point on the vertical line . Choose point on the horizontal line . Step 2 · Calculate the Lengths of the Three Sides Since is a vertical line segment along : Since is a horizontal line segment along : Since , find the hypotenuse using Pythagoras theorem: Answer > For vertices , , and : (Answers may vary depending on the chosen vertices and ) Common Mistakes Quadrant Placement Error: Plotting in Quadrant II instead of Quadrant IV . Non-perpendicular Sides: Choosing arbitrary points for and without ensuring two sides are along orthogonal lines, which leads to a triangle that is not right-angled. Negative Lengths: Forgetting to take the absolute value when subtracting coordinates to find side lengths.

Orienting Yourself: The Use of Coordinates | EOT

Question 4

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.)

Solution

Understand the Question

To plot $Z(5, -6)$ , move $5$ units right along the positive $x$ -axis and $6$ units down along the negative $y$ -axis (Quadrant IV).
To form a right-angled triangle $\Delta IZN$ , choose two other points $I$ and $N$ such that one side is vertical and one side is horizontal, making the angle between them $90^\circ$ .
The lengths of the sides can be determined using coordinate differences and the Pythagoras theorem.
Note: Different choices for points $I$ and $N$ are valid, so side lengths may vary.

Step 1 · Plot Point Z and Choose Vertices I and N

Plot the point $Z(5, -6)$ in the fourth quadrant of the Cartesian coordinate system.To construct a right-angled triangle $\Delta IZN$ with $\angle Z = 90^\circ$ :

Choose point $I(5, -2)$ on the vertical line $x = 5$ .
Choose point $N(2, -6)$ on the horizontal line $y = -6$ .

Step 2 · Calculate the Lengths of the Three Sides

Since $IZ$ is a vertical line segment along $x = 5$ :

IZ = |-2 - (-6)| = |-2 + 6| = 4 \text{ units}

Since $ZN$ is a horizontal line segment along $y = -6$ :

ZN = |5 - 2| = 3 \text{ units}

Since $\angle IZN = 90^\circ$ , find the hypotenuse $IN$ using Pythagoras theorem:

\begin{aligned} IN^2 &= IZ^2 + ZN^2 \\ &= 4^2 + 3^2 \\ &= 16 + 9 \\ &= 25 \\ IN &= \sqrt{25} = 5 \text{ units} \end{aligned}

Answer

For vertices $Z(5, -6)$ , $I(5, -2)$ , and $N(2, -6)$ :

IZ = 4 \text{ units}, \quad ZN = 3 \text{ units}, \quad IN = 5 \text{ units}

(Answers may vary depending on the chosen vertices $I$ and $N$ )

Common Mistakes

Quadrant Placement Error: Plotting $Z(5, -6)$ in Quadrant II $(-5, 6)$ instead of Quadrant IV $(5, -6)$ .
Non-perpendicular Sides: Choosing arbitrary points for $I$ and $N$ without ensuring two sides are along orthogonal lines, which leads to a triangle that is not right-angled.
Negative Lengths: Forgetting to take the absolute value when subtracting coordinates to find side lengths.

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?