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Day 1 · Learn3 min read
Coordinate Geometry: Distance, Midpoint & Slope
Every point on the coordinate plane has an address: (x, y). Once two points have addresses, you can measure the line segment between them without a ruler — just arithmetic. Three tools do almost all the work: how far apart the points are, the middle point between them, and how tilted the line is.
The three formulas
For points (x₁, y₁) and (x₂, y₂), think of the horizontal gap (x₂ − x₁) and the vertical gap (y₂ − y₁).
Distance comes straight from the Pythagorean theorem: the two gaps are the legs of a right triangle, and the segment is the hypotenuse. Midpoint just averages the x's and averages the y's. Slope is "rise over run" — vertical change divided by horizontal change.
How to use them
Label your two points so you know which is (x₁, y₁).
Subtract in the same order for both x and y.
Keep signs — a negative slope means the line falls left to right.
Points A and B are plotted on the coordinate plane below. What is the distance AB?
Use the distance formula. The horizontal change is 1 − (−3) = 4 and the vertical change is 2 − (−1) = 3, so AB = √(4² + 3²) = √(16 + 9) = √25 = 5.
Q2
Segment AB has endpoints A(−4, 2) and B(2, −4), as shown. What are the coordinates of the midpoint of AB?
The midpoint is the average of the two endpoints: ((−4 + 2)/2, (2 + (−4))/2) = (−1, −1).
Q3
A line passes through points P(−2, −3) and Q(2, 3), as shown. What is the slope of the line?
Slope is rise over run: (3 − (−3)) / (2 − (−2)) = 6 / 4 = 3/2.
Q4
M(1, 0) is the midpoint of segment AB. One endpoint is A(−2, 3). What are the coordinates of endpoint B?
Because the midpoint is the average of the endpoints, B = 2M − A = (2·1 − (−2), 2·0 − 3) = (4, −3).
Q5
Triangle ABC has vertices A(−3, −1), B(1, −1), and C(1, 2). What is the perimeter of triangle ABC?
AB is horizontal with length |1 − (−3)| = 4; BC is vertical with length |2 − (−1)| = 3; and AC = √(4² + 3²) = 5. The perimeter is 4 + 3 + 5 = 12.
Ready for the real thing?
This was a taste of the fundamentals. Texas CBE exam prep is where you turn them into credit — full-length timed mock exams, and a review that tells you exactly which topics to go back to.