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Questions and Answers
What is the formula to find the midpoint of a line segment?
If you have two points (3, 4) and (7, 10), what is the midpoint of the line segment connecting them?
How can you calculate the slope of a line?
What does a slope of zero indicate about a line?
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If you have two points (5, 6) and (10, 6), what is the slope of the line passing through these points?
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What type of line has a slope with an absolute value greater than 1?
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In a coordinate plane, where is the origin located?
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What are the perpendicular number lines called in a coordinate plane?
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What do you call the point where both axes intersect on a coordinate plane?
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Study Notes
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.
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Description
Test your knowledge on the basics of coordinate geometry including the midpoint formula, slope calculation, and distance formula. Learn how to find midpoints, slopes, and distances between points on a coordinate plane.
