7.1 Distance, Midpoint & Slope

The Coordinate Toolkit for G-GPE

The New York Expressing Geometric Properties with Equations domain (standards GEO-G.GPE.4 through GEO-G.GPE.7) turns geometry into algebra. Every proof, partition, and area problem in this domain rests on three formulas: the distance formula, the midpoint formula, and the slope formula. None of the three is printed on the NYSED Next Generation Geometry Reference Sheet, so memorize them cold before test day. Coordinate geometry carries 12-18% of the Regents blueprint, and these tools show up both in Part I multiple choice and inside the longer Part II-IV constructed-response proofs.

Formula table

Distance Formula

The distance formula is simply the Pythagorean theorem applied to the right triangle whose legs are the horizontal change Δx and vertical change Δy between two points.

Worked example: find the distance between A(1, −2) and B(7, 6). The horizontal change is 7 − 1 = 6; the vertical change is 6 − (−2) = 8. Then d = √(6² + 8²) = √(36 + 64) = √100 = 10. Recognizing the 6-8-10 pattern (a scaled 3-4-5 triple) saves calculator time.

Two shortcuts:

• Horizontal segment (equal y-values): distance = |x₂ − x₁|. For A(−2, 5) and B(4, 5), AB = |4 − (−2)| = 6.
• Vertical segment (equal x-values): distance = |y₂ − y₁|.

Because every term is squared, the order of subtraction never changes the answer — but you must square the difference, not subtract the squares of the coordinates.

Midpoint Formula

The midpoint is the average of the two x-coordinates and the average of the two y-coordinates. For endpoints (2, 3) and (8, 11): M = ((2 + 8)/2, (3 + 11)/2) = (5, 7).

A favorite Regents twist runs the formula backward: you are given one endpoint and the midpoint and must find the other endpoint. If A(2, 3) and midpoint M(5, 7), then apply B = 2M − A to each coordinate: B = (2·5 − 2, 2·7 − 3) = (8, 11). Do not average the two numbers you already know — double the midpoint and subtract the known endpoint.

Slope Formula

Slope measures rise over run: m = (y₂ − y₁)/(x₂ − x₁). For (0, 2) and (3, 8), m = (8 − 2)/(3 − 0) = 6/3 = 2. Read the sign and the special cases carefully:

• Positive slope rises from left to right; negative slope falls.
• Horizontal line: slope = 0 (the numerator is 0).
• Vertical line: slope is undefined (the denominator is 0) — never call it "zero."

The most common error is mixing the order: subtract the coordinates in the same order in both numerator and denominator. Subtracting y in one order and x in the other flips the sign of the slope.

Classifying Segments and Triangles

The three tools let you name a figure from its coordinates alone:

• Distance compares side lengths. A triangle with three equal sides is equilateral, exactly two equal is isosceles, and none equal is scalene.
• Slope detects right angles and parallel sides. If two sides have slopes whose product is −1, they meet at a right angle.

Worked classification using the midsegment: In triangle ABC with A(−4, 2), B(8, 6), C(2, −6), let M be the midpoint of AB and N the midpoint of AC. Then M = (2, 4) and N = (−1, −2). Distance MN = √((2 − (−1))² + (4 − (−2))²) = √(3² + 6²) = √45 = 3√5. The Triangle Midsegment Theorem predicts MN is half of BC; indeed BC = √(6² + 12²) = √180 = 6√5, so MN = 3√5 confirms the tools agree.

Worked classification — right triangle: Take P(1, 1), Q(5, 1), R(1, 4). Then PQ = |5 − 1| = 4 (horizontal) and PR = |4 − 1| = 3 (vertical), so PQ meets PR at a right angle, and QR = √(4² + 3²) = 5. Because no two sides are equal, PQR is a scalene right triangle. Move R to (1, 5) and PR becomes 4, matching PQ, which would make PQR an isosceles right triangle instead.

Collinearity and the Order of Subtraction

Three points are collinear (on one straight line) when the slope between the first pair equals the slope between the second pair. For A(1, 2), B(3, 6), C(5, 10): slope AB = (6 − 2)/(3 − 1) = 2 and slope BC = (10 − 6)/(5 − 3) = 2, so A, B, and C lie on one line. This is exactly the slope-matching test used to prove sides parallel in Section 7.2.

One more note on order: for (4, 9) and (1, 3), slope = (9 − 3)/(4 − 1) = 6/3 = 2, and reversing both endpoints to (3 − 9)/(1 − 4) = (−6)/(−3) = 2 gives the same result. A graphing calculator is allowed on the Regents, but these three formulas are usually faster by hand.

Common traps

• Squaring individual coordinates instead of the differences inside the distance formula.
• Averaging only one coordinate in the midpoint — both must be averaged.
• Calling a vertical line's slope "0" when it is actually undefined.
• Leaving √45 unsimplified when the answer choices show the simplified form 3√5.