Geometry

Coordinate Proofs

In geometry, you can prove things with logical arguments or with coordinates and algebra. Coordinate proofs put shapes on a grid and use formulas — slope, distance, midpoint — to verify properties like parallel sides, equal lengths, or right angles. The best part? You can see the proof update as you move the points.

1. Proving Lines Are Parallel (Equal Slopes)

Two lines are parallel if and only if they have the same slope. Let’s place two lines on the coordinate plane and check. Move the sliders to change the slopes and watch whether the lines stay parallel.

Slope of line 1 (m)1

-33

Intercept of line 12

-55

Slope of line 2 (m)1

-33

Intercept of line 2-1

-55

\text{Line 1: } y = 1x + 2 \quad \text{Line 2: } y = 1x + -1

\text{Slopes equal? } 1 = 1

Try This

Set both slopes to the same value (e.g., m = 1.5 for both). The lines never cross — they’re parallel! Now change one slope slightly. The lines immediately cross somewhere. Parallel means identical slopes, period. That’s the coordinate proof for parallelism.

2. Proving Lines Are Perpendicular (Slopes Multiply to -1)

Two lines are perpendicular (meet at a 90-degree angle) when the product of their slopes equals -1. Another way to say it: their slopes are negative reciprocals.

Slope of line 12

0.24

Intercept 10

-55

\text{Line 1 slope: } 2 \quad \text{Line 2 slope: } \frac{-1}2

\text{Product: } 2 \times \frac{-1}2 = -1 \; \checkmark

Connection

No matter what slope you pick for Line 1, Line 2 is always perpendicular because its slope is forced to be -1/m. The product is always -1. This is the coordinate proof for perpendicularity: compute both slopes, multiply them, and check if you get -1.

3. Distance Formula: Proving Equal Side Lengths

To prove a triangle is isosceles or equilateral, you need the distance formula: d = sqrt((x2-x1)^2 + (y2-y1)^2). Let’s place a triangle and measure its sides.

Point A: x0

-55

Point A: y4

-55

Point B: x-3

-55

Point B: y-2

-55

A = (0, 4), \quad B = (-3, -2), \quad C = (3, -2)

d(A,B) = \sqrt{(-3-0)^2 + (-2-4)^2}

Try This

Try to make the triangle isosceles by adjusting points A and B so that two sides have equal length. For example, set A = (0, 4) and B = (-3, -2). Now compute d(A,C) and d(B,C) with C = (3, -2). If they’re equal, you’ve proven isosceles using coordinates!

4. Midpoint Formula: Proving Bisection

The midpoint of a segment with endpoints (x1, y1) and (x2, y2) is ((x1+x2)/2, (y1+y2)/2). To prove that a point bisects a segment, show it’s the midpoint.

P1: x-4

-66

P1: y-2

-66

P2: x4

-66

P2: y3

-66

P_1 = (-4, -2), \quad P_2 = (4, 3)

\text0 = \left(\frac{-4+4}0, \frac{-2+3}0\right)

Connection

The yellow horizontal line shows the y-coordinate of the midpoint. It always passes through the exact center of the segment. To prove bisection in a coordinate proof: compute the midpoint of the segment, then show that the bisector passes through that point.

5. Putting It All Together: Proving a Parallelogram

A quadrilateral is a parallelogram if opposite sides are parallel (equal slopes). Let’s set up a quadrilateral and verify.

Horizontal offset3

Vertical offset2

-35

\text{Vertices: } A(0,0),\; B(5,0),\; C(5+3,2),\; D(3,2)

\text{Slope AB} = 0, \quad \text{Slope DC} = \frac{2-2}{(5+3)-3} = 0 \;\checkmark

\text{Slope AD} = \frac{2}3, \quad \text{Slope BC} = \frac{2}3 \;\checkmark

Challenge

Challenge: AB is always horizontal (slope 0), and DC is always parallel to it. The sliders control where D goes, and C follows to maintain the parallelogram shape. No matter what values you pick, opposite sides always have equal slopes. That’s the coordinate proof: show that both pairs of opposite sides have equal slopes, therefore the quadrilateral is a parallelogram.

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