Coordinate (analytic) geometry
The idea
Coordinate geometry lets algebra prove geometric claims: place a figure on the plane and every property becomes a computation. Three formulas do most of the work, and each flows from ideas you already hold — the distance formula √((x₂ − x₁)² + (y₂ − y₁)²) is the Pythagorean theorem dressed in coordinates, the midpoint formula averages the coordinates, and slope measures direction, with parallel lines sharing slopes and perpendicular lines having slopes that multiply to −1. Claims like 'this triangle is right-angled' become checkable arithmetic.
A coordinate proof has a rhythm: compute the relevant distances or slopes, compare them, and translate the comparison back into geometric language. Two independent routes to the same conclusion — verifying a right angle both by perpendicular slopes and by the Pythagorean converse — turn a calculation into a convincing argument. The misconception to evict is that opposite slopes mean perpendicular lines; the true test is opposite RECIPROCALS, like 4/3 and −3/4, whose product is −1. Slopes of 2 and −2 are merely mirror-steep, not perpendicular.
Worked example
Triangle PQR has vertices P(1, 1), Q(4, 5), and R(8, 2). Show that it is an isosceles right triangle and find its area.
- Compute the three side lengths with the distance formula: PQ = √((4 − 1)² + (5 − 1)²) = √(9 + 16) = 5, QR = √((8 − 4)² + (2 − 5)²) = √(16 + 9) = 5, and PR = √((8 − 1)² + (2 − 1)²) = √(49 + 1) = √50.
- Two equal sides (PQ = QR = 5) make the triangle isosceles. For the right angle, test the Pythagorean converse: PQ² + QR² = 25 + 25 = 50 = PR², so the angle between PQ and QR — the angle at Q — is right.
- Confirm by slopes as an independent route: slope of PQ = (5 − 1)/(4 − 1) = 4/3 and slope of QR = (2 − 5)/(8 − 4) = −3/4. Their product is −1, so the segments are perpendicular at Q.
- The two legs are the perpendicular sides, so area = (1/2) × 5 × 5 = 12.5 square units. As a final plausibility check, the hypotenuse √50 ≈ 7.07 is indeed the longest side.
Answer. Triangle PQR is an isosceles right triangle with its right angle at Q, and its area is 12.5 square units.
Check your understanding
- How is the distance formula the Pythagorean theorem wearing coordinates, and which right triangle does it secretly draw?
- Why does the product of the slopes of perpendicular lines equal −1, and where does that rule break down for vertical lines?
- What is gained by verifying the right angle through two different methods within one coordinate proof?
- How would you choose convenient coordinates to prove a general fact, such as the diagonals of a rectangle being equal?
Build the foundations first
Coordinate (analytic) geometry builds on these concepts. If any feel shaky, start there.