Co-ordinate Geometry — Competency Focused Questions

The coordinates of a point are (2, 8). Its distance from x-axis is :

1. 2 units

2. 8 units

3. (2 + 8) units

4. $\sqrt{2^2 + 8^2}$ units

Answer

Point = (2, 8)

Distance from x-axis = 8 units

Hence, option 2 is the correct option.

A point is 4 units away from the origin. If it lies in the first quadrant, how many such points exist?

1. One

2. Two

3. Four

4. Infinitely many

Answer

A point is 4 units from the origin.

⇒ All points lie on a circle of radius 4.

⇒ In the first quadrant, there are infinitely many points on that arc.

Hence, option 4 is the correct option.

The coordinates of two points A and B are (5, 7) and (5, -5). The length of the line segment AB is 7 - (-5) units. This statement is :

1. true

2. false

3. can't say

4. none of these

Answer

According to question :

Length of AB = 7 - (-5) = 7 + 5 = 12.

We will find the length of AB by using distance formula,

$AB = \sqrt{(5 - 5)^2 + (-5 -7)^2}\\[1em] = \sqrt{0 + (-12)^2}\\[1em] = \sqrt{144} \\[1em] = 12 \text{ units}.$

Hence, option 1 is the correct option.

Which of the following points does not lie in any quadrant?

1. (1, 2)

2. (4, 4)

3. (-2, -1)

4. (0, 5)

Answer

As the point (0, 5) lies on y-axis.

⇒ It does not lie in any quadrant.

Hence, option 4 is the correct option.

A point on the coordinate plane is 3 units far from x-axis and 4 units far from y-axis. Which of the following can be the coordinates of the point?

1. (3, 4)

2. (-3, -4)

3. both (a) and (b)

4. none of these

Answer

As, distance from x-axis is 3 units.

⇒ |y| = 3

Distance from y-axis is 4 units.

⇒ |x| = 4

Thus, the points can be (4, 3), (4, -3), (-4, 3), (-4, -3).

Hence, option 4 is the correct option.

Which of the following points is nearest to x-axis?

1. (-1, 4)

2. (4, -3)

3. (2, 2)

4. (1, -4)

Answer

Distance from x-axis = Absolute value of y coordinate

Smallest distance = 2

⇒ (2, 2) is the nearest point to x-axis.

Hence, option 3 is the correct option.

A circle is drawn with origin (O) as centre and radius 4 units. Among the points (4, 2), (2, 2), (-5, 1), (-1, -2), (-4, 1), (1, 1), (-2, 4), state which lie inside the circle and which lie outside the circle.

Answer

Centre = O(0, 0)

Radius = 4

Equation of circle :

x² + y² = radius²

x² + y² = 4²

x² + y² = 16

If point lies inside the circle,

Equation will be :

x² + y² < 16

If point lies outside the circle,

Equation will be :

x² + y² > 16

Now, we will check for each point.

1. (4, 2)

4² + 2² = 16 + 4 = 20

20 > 16

So, point is outside the circle.

2. (2, 2)

2² + 2² = 4 + 4 = 8

8 < 16

So, point is inside the circle.

3. (-5, 1)

(-5)² + 1² = 25 + 1 = 26

26 > 16

So, point is outside the circle.

4. (-1, -2)

(-1)² + (-2)² = 1 + 4 = 5

5 < 16

So, point is inside the circle.

5. (-4, 1)

(-4)² + 1² = 16 + 1 = 17

17 > 16

So, point is outside the circle.

6. (1, 1)

1² + 1² = 1 + 1 = 2

2 < 16

So, point is inside the circle.

7. (-2, 4)

(-2)² + 4² = 4 + 16 = 20

20 > 16

So, point is outside the circle.

Hence, points lying

Inside the circle :(2, 2), (-1, -2), (1, 1)

Outside the circle : (4, 2), (-5, 1), (-4, 1), (-2, 4)

A right angled triangle ABC is drawn on the coordinate plane as shown below. Find the area of the right triangle.

Answer

From graph,

A = (-2, 5) and B = (7, -3)

As,

x-coordinate of point C = x-coordinate of point A

y-coordinate of point C = y-coordinate of point B

Coordinates of C = (-2, -3)

Now,

Area of right triangle = $\dfrac{1}{2} \times \text {base} \times \text{height}$

We will find base and height by using distance formula,

Distance between two points = $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Base = CB

$CB =\sqrt{(7 - (-2))^2 + (-3 - (-3))^2}\\[1em] =\sqrt{(7 + 2)^2 + (-3 + 3)^2}\\[1em] =\sqrt{(9)^2 + (0)^2}\\[1em] =\sqrt{81}\\[1em] = 9.$

Base = 9 units

Height = AC

$AC =\sqrt{(-2 - (-2))^2 + (-3 - 5)^2}\\[1em] =\sqrt{(-2 + 2)^2 + (-3 - 5)^2}\\[1em] =\sqrt{(0)^2 + (-8)^2}\\[1em] =\sqrt{64}\\[1em] = 8.$

Height = 8 units

Area of right triangle ABC = $\dfrac{1}{2} \times \text {base} \times \text{height}$

= $\dfrac{1}{2} \times 9 \times 8$

= 36 sq. units

Hence, area of triangle = 36 sq. units.