Algebra and Geometry

Did you know Algebra and Geometry go hand-in-hand?

graph with point (12,5)
Here the point (12,5) is
12 units along, and 5 units up

Cartesian Coordinates can be used to locate a point by how far along and how far up it is:

We can then place shapes like triangles and quadrilaterals onto the coordinate plane and use the power of algebra to discover things about them.

Imagine just one side of a shape, it has two points that make a line segment:

point 1 at (x₁,y₁) and point 2 at (x₂,y₂)

Now we can calculate:

This tells us how long a line segment is. It is actually just the Pythagorean Theorem (a² + b² = c²) in disguise!

Distance = √[(x₂ − x₁)² + (y₂ − y₁)²]

This finds the point that's exactly in the middle of a line segment. It is the average of the x-coordinates and the average of the y-coordinates.

Midpoint = ( x₁ + x₂2, y₁ + y₂2 )

This measures the steepness (gradient) of a line by looking at the "Rise over Run."

Slope (m) =

y₂ − y₁x₂ − x₁

Parallel Lines have the same slope: m₁ = m₂
Perpendicular Lines (lines meeting at a 90° right angle) have slopes that multiply to give −1: m₁ × m₂ = −1

For Triangles

Here are some ways we can use them with triangles:

Isosceles triangle: Use the Distance Formula on all three sides. When two sides have equal lengths, it's isosceles
Equilateral triangle: Use the Distance Formula. When all three sides are equal, it's equilateral
Right triangle: Use the Slope Formula. When two sides have slopes that multiply to equal −1, they meet at a right angle

The coordinates are A(1, 1), B(4, 5), and C(8, 2).

Now we multiply them together: (4/3) × (−3/4) = −12/12 = −1

Because the product of the slopes is −1, side AB is perpendicular to side BC.

So, ΔABC is a right triangle!

Quadrilaterals (4-sided shapes) love to hide their true identities. We can use our formulas to unmask them:

Property: Opposite sides are parallel
How to check: Find the Slope of all 4 sides. Opposite sides must have the same slope

Property: It's a parallelogram with 90° corners
How to check: Check if adjacent slopes multiply to −1, or use the Distance Formula to show the two diagonals are equal in length

Property: It's a parallelogram with 4 equal sides
How to check: Use the Distance Formula to show all four sides are equal, or use Slope to prove the diagonals are perpendicular

Step 1: Check Slopes (are sides parallel?)
Slope QR = 1/3 and Slope PS = 1/3 (Parallel!)
Slope PQ = −3 and Slope RS = −3 (Parallel!)

Step 2: Check for Right Angles
Slope PQ (−3) × Slope QR (1/3) = −1. (Right angles!)

Step 3: Check Side Lengths
Length PQ = √10 and Length QR = √10. (Equal sides!)

Conclusion: Because opposite sides are parallel, corners are 90°, and sides are equal, it is a Square!

For you: plot it and see.

And this is just the beginning of how Algebra and Geometry can work together.