Coordinate Geometry Basics Quiz

Coordinate Geometry

Coordinate geometry is a mathematical field that allows us to represent geometric figures on a coordinate plane using numerical coordinates. This plane consists of two perpendicular number lines called axes, with the horizontal axis being the x-axis and the vertical axis being the y-axis. The origin, where both axes intersect, is marked as (0, 0).

Midpoint Formula

To find the midpoint of a line segment connecting points ((x_1, y_1)) and ((x_2, y_2)), you can use the following formula:

[ \text{Midpoint} = \left(\frac{x_2 + x_1}{2}, \frac{y_2 + y_1}{2}\right) ]

For instance, if you want to find the midpoint of a segment with (x_1 = 10), (y_1 = 6), (x_2 = 2), and (y_2 = 8), you would plug these values into the formula:

[ \text{Midpoint} = \left(\frac{10 + 2}{2}, \frac{6 + 8}{2}\right) = \left(\frac{12}{2}, \frac{14}{2}\right) = (6, 7) ]

So, the midpoint of this segment is ((6, 7)).

Slope of a Line

The slope of a line can be calculated using the following formula:

[ m = \frac{y_2 - y_1}{x_2 - x_1} ]

where ((x_1, y_1)) and ((x_2, y_2)) are any two points on the line. The slope indicates the steepness or inclination of the line and is often represented as a fraction or a decimal value. If the slope is zero, the line is horizontal; if the absolute value of the slope is 1, the line is a unit slope line parallel to the axes; and if the absolute value of the slope is greater than 1, the line gets steeper.

Distance Formula

The distance between two points ((x_1, y_1)) and ((x_2, y_2)) on the coordinate plane can be calculated using the distance formula:

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

For example, let's find the distance between points (A(3, 5)) and (B(10, 8)):

[ d = \sqrt{(10 - 3)^2 + (8 - 5)^2} = \sqrt{49 + 9} = \sqrt{58} ]

So, the distance between point (A) and (B) is approximately 8 units.