Cartesian Plane Practice Worksheet

Coordinate Geometry is a fundamental part of Mathematics that is used to solve problems related to the plotting of various curves on the graphs.

In this article, we will make use of algebra to solve geometrical problems. A Cartesian Plane is a system in which real numbers are represented by points. A Cartesian plane is usually n-dimensional. But we usually consider 2D or 3D planes.

• Coordinates: The points plotted on the plane.
• Axes: The dimensions of the plane.
• Abscissa: The x coordinate of the plane.
• Ordinate: The y coordinate of the plane.

Cartesian Plane Definition

Cartesian Plane has 2 dimensions or axes. The axes are defined by X and Y and the coordinates are denoted by(x,y). Usually, the axes intersect at a particular point known as the origin. The axes are perpendicular to one another. These axes divide the plane into 4 parts and these 4 parts are known as Quadrants.

• Quadrant I: Both coordinates are positive.
• Quadrant II: Here x is negative, y is positive.
• Quadrant III: Both x and y coordinates are negative.
• Quadrant IV: In this quadrant x is positive, y is negative.

Formulas related to 2D Cartesian Plane

A straight line passing through coordinates (x, y) where m is the slope of the equation, c is the y intercept. This formula is also called Slope Intercept form.

y = mx + c

For calculating the distance between two points denoted by (a,b) and (c,d), we will use the Euclidean distance which is as follows:

s=\sqrt{(c-a) ^2+(d-b) ^2 }

To find slope or the tangent of the line in which the the line passes through points (a, b) and (c, d) we will use the formula:

m = (d - b)/(c - a)

To find the midpoint of the line passing through two points (a,b) and (c,d), the coordinate is as follows:

x'=\frac{(a+c)}{2} \\y'=\frac{(b+d)}{2}

Let there be a line whose slope is m and passing through point (a,b). The equation of the line is as follows:

y - b = m(x - a)

Formulas related to 3D Cartesian Plane

The formulas related to 3D Plane

Equation of Plane: A plane in 3D space can be represented by the equation:

Ax + By + Cz = D

where A, B, C, and D are constants.

Finding Distance: The distance between two points (x_1,y_1,z_1) and (x_2,y_2,z_2)can be found by refining the Euclidean Distance formula

d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}

Midpoint Formula: The midpoint between two points (x_1,y_1,z_1) and (x_2,y_2,z_2) can be found

x'=\frac{(x_1+x_2)}{2} \\y'=\frac{(y_1+y_2)}{2} \\z'=\frac{(z_1+z_2)}{2}

Worksheet- Cartesian Plane

You can download the pdf of the worksheet and solutions of the question in worksheet from- Worksheet Cartesian Plane

Cartesian Plane: Practice Questions with Solution

Question 1: Which points lie in Second Quadrant (2, 4), (-3, 3), (-6, 5)

Solution:

In second quadrant we know that value of X coordinate is negative and value of Y coordinate is positive

Consider the first point (2, 4). Here both the real numbers are positive. So it lies in first quadrant

For (-3, 3) the abscissa is negative and ordinate is positive. So it lies in second quadrant.

For (-6, 5) , abscissa is negative while ordinate is positive. So it lies in second quadrant.

Question 2: Identify the quadrants in which the points lie (-6, 4), (4, 6), (6, -4), (-6, -4)

Solution:

• (-6, 4): Second quadrant
• (4, 6): First Quadrant
• (6, -4): Fourth Quadrant
• (-6, -4): Third Quadrant

Question 3: Find the equation of the line whose slope has value -5 and the y-intercept is 7

Solution:

We know that the equation of the line with slope m and y intercept b is given by y = mx + c

Here m = -5 and c = 7

y = (-5)x + 7

Question 4: Find the equation of the plane where values of the constants: A = 6, B = 7, C = 9, D = 10

Solution:

Equation of the line is also given by Ax + By + Cz = D

Putting the values as mentioned the equation is:

6x + 7y + 9z = 10

Question 5: Find the slope of the equation that passes through points (6, 5) and (5, 2)

Solution:

Slope formula is m=\frac{(d-b)}{(c-a)}

m = \frac{(2-5)}{(5-6)} \\= 3

Question 6: Use the Euclidean distance formula to find the distance between two points (2, 3) and (-4, 5)

Solution:

Euclidean distance is given by:

d=\sqrt{((5-3) ^2+(-4-2) ^2) } \\=\sqrt{(4+36) } \\=\sqrt{40}

Question 7: Use a graph paper and plot the points (3, 4) , (-4, 5), (-6, -7)

Solution:

Below is the solution for the same

Question 8: Find the midpoint of the line passing through the points (-3, -2) and (-9, -5)

Solution:

Formula for the midpoint to pass through points (a, b) and (c, d) is given by

x=\frac{(a+c)}{2} \\y=\frac{(b+d)}{2}

The midpoints

x=\frac{(-3-9)}{2}= -6 \\y=\frac{(-2-5)}{2} =-3.5

Therefore the midpoint is (-6,-3.5)

Question 9: Below is the figure with some coordinates. Identify the values and identify quadrants as well

Solution:

We can see one point has abscissa 2 and ordinate is 4. The first coordinate lies in first quadrant

Second point is (-4,-5) and it lies in fourth quadrant.

Question 10: Given the points A(1,2) and B(x,6) in a 2-dimensional space. The midpoint is (0,4) find the value of x.

Solution:

Using the midpoint formula we get

0=\frac{(x+1)}{2} \\=>x=-1

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• Cartesian Plane
• Cartesian Coordinate System
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