Geometry • Topic 18

Cartesian Coordinates

Coordinate geometry (analytic geometry) translates geometric problems into algebraic ones. It is particularly effective for problems involving loci, distances, or specific ratios.

Fundamental Formulas

For points $A (x_{1}, y_{1})$ and $B (x_{2}, y_{2})$ :

Distance Formula:

d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}

Midpoint Formula:

M = (\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2})

Slope ( $m$ ):

m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}

Parallel lines have equal slopes ( $m_{1} = m_{2}$ ).

Perpendicular lines have negative reciprocal slopes ( $m_{1} m_{2} = - 1$ ).

The Shoelace Formula

A powerful algorithm to find the area of a polygon given the coordinates of its vertices

(x_{1}, y_{1}), \dots, (x_{n}, y_{n})

in order.

Theorem (Shoelace Formula)

A re a = \frac{1}{2} ∣ (x_{1} y_{2} + x_{2} y_{3} + \dots + x_{n} y_{1}) - (y_{1} x_{2} + y_{2} x_{3} + \dots + y_{n} x_{1}) ∣

[Image of Shoelace Formula Diagram]

Mnemonic: List coordinates in a column, repeating the first at the bottom. Multiply diagonals down-right (add) and up-right (subtract).

Equation of a Circle

A circle with center

(h, k)

and radius

r

(x - h)^{2} + (y - k)^{2} = r^{2}

Practice Problems

Exercise (Problem 1)

Find the area of the pentagon with vertices

(0, 0), (2, 1), (3, 3), (1, 4), (- 1, 2)

using the Shoelace formula.

Exercise (Problem 2)

Prove that the quadrilateral formed by the midpoints of any quadrilateral is a parallelogram (Varignon's Theorem) using coordinates.

Exercise (Problem 3)

Find the locus of points

P

such that

P A^{2} + P B^{2} = k

for fixed points

A, B

. (Hint: Set

A = (- a, 0), B = (a, 0)