Coordinate geometry: distance, midpoint, and slope proofs (Geometry) · lesson

The distance formula

💡 Distance is the Pythagorean theorem on a grid: d = √((x₂ − x₁)² + (y₂ − y₁)²).

Between two points, the change in x and the change in y are the legs of a right triangle. The distance is its hypotenuse.

Subtract the x-values, subtract the y-values, square both, add, and take the square root. Watch the signs when you subtract a negative.

For a perimeter, find each side's length and add them up.

🧠 Let's work it out

Find the distance from (1, 1) to (9, 16).

Change in x: 9 − 1 = 8.
Change in y: 16 − 1 = 15.
Square and add: 8² + 15² = 64 + 225 = 289.
Take the root: √289 = 17.

What do you think comes next?

coordinates: the pair of numbers (x, y) that locate a point

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Midpoints and partway points

💡 To go a fraction of the way from A to B, take that fraction of the step from A to B and add it to A.

The midpoint is halfway, so average the x-values and average the y-values.

If you know one endpoint and the midpoint, the step from the endpoint to the midpoint is half the trip. Take the same step again to reach the other end.

For a point 1/4 or 1/3 of the way, find the full step from A to B. Multiply it by the fraction. Add that to A.

🧠 Let's work it out

M(1, 2) is the midpoint of AB, and A is (−3, 5). Find B.

The step from A to M is (1 − (−3), 2 − 5) = (4, −3).
Take that step again from M: (1 + 4, 2 − 3).
So B = (5, −1).

What do you think comes next?

midpoint: the point exactly halfway between two points
partition: to split a segment into parts with a given ratio

Slopes of parallel and perpendicular lines

💡 Parallel lines have equal slopes; perpendicular slopes are negative reciprocals that multiply to −1.

Slope = change in y ÷ change in x. If a line is written like 2x + 5y = 10, solve for y first. The number in front of x is the slope.

Parallel lines have the same slope. Perpendicular lines have slopes that flip and change sign: 2 pairs with −1/2.

To write a line through a point, put the slope and the point into y = mx + b and solve for b.

🧠 Let's work it out

Write the line perpendicular to y = 2x + 5 through (4, 1).

Flip 2 and change the sign: the new slope is −1/2, or −0.5.
Put in the point: 1 = −0.5(4) + b.
So 1 = −2 + b, and b = 3.
The line is y = −0.5x + 3.

What do you think comes next?

slope: how steep a line is: change in y ÷ change in x
negative reciprocal: flip a fraction and change its sign: 2/3 becomes −3/2
slope-intercept form: y = mx + b, where m is the slope and b is where the line crosses the y-axis

Proving what a shape is

💡 Use slopes to test for parallel and perpendicular sides, and distances to test for equal sides.

Find the slope and length of every side. Two pairs of parallel sides make a parallelogram.

If all four sides are equal, it is a rhombus. If the corners are right angles (side slopes multiply to −1), it is a rectangle. If both are true, it is a square.

🧠 Let's work it out

What shape is ABCD, with A(0, 0), B(2, 1), C(1, 3), D(−1, 2)?

Slopes: AB = 1/2, BC = −2, CD = 1/2, DA = −2.
Opposite sides have equal slopes, so they are parallel.
1/2 × (−2) = −1, so the corners are right angles.
Every side is √5 long, so all sides are equal.
Right angles and equal sides make a square.

What do you think comes next?

parallelogram: a four-sided shape with two pairs of parallel sides
rhombus: a four-sided shape with all sides equal
quadrilateral: any shape with four sides