Math Core

When a synthetic idea doesn't come, coordinates always give you a way in: put the figure on a grid, name the points, and turn geometry into algebra. On the AMC the trick is to do this efficiently: choose axes that make the numbers small, and know a handful of formulas (shoelace, point-to-line distance, reflections) so the algebra stays short.

The basic formulas

For points $(x_1, y_1)$ and $(x_2, y_2)$ :

Distance: $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ .
Midpoint: $\left(\dfrac{x_1 + x_2}{2}, \dfrac{y_1 + y_2}{2}\right)$ .
Slope: $m = \dfrac{y_2 - y_1}{x_2 - x_1}$ . Parallel lines have equal slopes; perpendicular lines have slopes whose product is $-1$ .
Circle with center $(h, k)$ and radius $r$ : $(x - h)^2 + (y - k)^2 = r^2$ . Complete the square to find the center and radius of $x^2 + y^2 + Dx + Ey + F = 0$ .

Two power tools

The shoelace formula

A polygon with vertices $(x_1, y_1), (x_2, y_2), \dots, (x_n, y_n)$ listed in order around the boundary has area

$\frac12 \left| (x_1 y_2 - x_2 y_1) + (x_2 y_3 - x_3 y_2) + \dots + (x_n y_1 - x_1 y_n) \right|.$

Each term $\tfrac12(x_i y_{i+1} - x_{i+1} y_i)$ is the signed area of the triangle formed by the origin and one edge. Adding them around the polygon, the parts outside the polygon cancel.

Distance from a point to a line

The distance from $(x_0, y_0)$ to the line $ax + by + c = 0$ is

$\frac{|a x_0 + b y_0 + c|}{\sqrt{a^2 + b^2}}.$

The quadrilateral with vertices (0, 0), (6, 1), (5, 6) and (1, 4).Open in grapher →

Worked example: Shoelace

Find the area of the quadrilateral with vertices $(0, 0)$ , $(6, 1)$ , $(5, 6)$ , $(1, 4)$ .

\begin{aligned} x_1 y_2 - x_2 y_1 &= 0 \cdot 1 - 6 \cdot 0 = 0 \\ x_2 y_3 - x_3 y_2 &= 6 \cdot 6 - 5 \cdot 1 = 31 \\ x_3 y_4 - x_4 y_3 &= 5 \cdot 4 - 1 \cdot 6 = 14 \\ x_4 y_1 - x_1 y_4 &= 1 \cdot 0 - 0 \cdot 4 = 0 \end{aligned}

The area is $\tfrac12(0 + 31 + 14 + 0) = \dfrac{45}{2}$ .

Common mistake

List the vertices in order around the polygon (either direction). If you list them in a crossing order, the shoelace formula quietly gives a wrong answer.

Reflections for shortest paths

To find the shortest path from $A$ to a line and then to $B$ (with $A$ and $B$ on the same side), reflect $B$ over the line to $B'$ . Every path $A \to P \to B$ has the same length as $A \to P \to B'$ , and the shortest of those is the straight segment $AB'$ .

Reflect (1, 3) to (1, -3). The straight segment from (1, -3) to (7, 5) crosses the x-axis at the best point P.Open in grapher →

Worked example: Shortest path via reflection

Find the length of the shortest path from $A = (1, 3)$ to a point $P$ on the $x$ -axis and then to $B = (7, 5)$ .

Reflect $A$ to $A' = (1, -3)$ . Then $AP = A'P$ , so the shortest path has length $A'B = \sqrt{6^2 + 8^2} = 10$ . (It hits the axis at $P = \left(\tfrac{13}{4}, 0\right)$ .)

Choosing good coordinates

Worked example: Coordinates in a square

Square $ABCD$ has side $6$ . Point $M$ is the midpoint of $\overline{BC}$ , and $\overline{DM}$ meets diagonal $\overline{AC}$ at $P$ . Find the area of $\triangle APD$ .

Place $A = (0, 0)$ , $B = (6, 0)$ , $C = (6, 6)$ , $D = (0, 6)$ , so $M = (6, 3)$ . Line $AC$ is $y = x$ ; line $DM$ is $y = 6 - \tfrac{x}{2}$ . They meet where $x = 6 - \tfrac{x}{2}$ , so $P = (4, 4)$ . Triangle $APD$ has base $AD = 6$ (on the $y$ -axis) and height $4$ (the $x$ -coordinate of $P$ ), so its area is $12$ .

Tip

Put a vertex at the origin, and a side or axis of symmetry along an axis. Use zeros generously: every zero coordinate removes terms from every formula you use later.

Practice

Practice 1

The lines $y = 2x + 1$ and $y = -x + 7$ meet at point $P$ . How far is $P$ from the origin?

Practice 2

What is the area of the triangle with vertices $(1, 2)$ , $(7, 4)$ and $(3, 10)$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 3

What is the $y$ -intercept of the perpendicular bisector of the segment joining $(2, 1)$ and $(8, 5)$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 4

A circle is tangent to both of the lines $3x + 4y = 7$ and $3x + 4y = -8$ . What is the area of the circle?

Practice 5

Point $P$ moves along the $x$ -axis. What is the least possible value of $AP + PB$ , where $A = (2, 5)$ and $B = (10, 3)$ ?

Practice 6

How many points with integer coordinates lie on the segment from $(3, 7)$ to $(51, 43)$ , including the endpoints?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 7

What is the area of the region of points $(x, y)$ with $|x| + |y| \le 6$ and $y \le 2$ ?

Practice 8

The line $y = x$ meets the circle $x^2 + y^2 - 6x + 8y = 0$ in a chord. How long is the chord?

Practice 9

A line with negative slope passes through $(4, 3)$ and forms a triangle with the positive $x$ - and $y$ -axes. What is the smallest possible area of this triangle?

Real contest practice

2019 AMC 10A, Problem 7: the area of a triangle bounded by three lines, found from intersection points.
2020 AMC 12A, Problem 17: a quadrilateral with vertices on $y = \ln x$ ; the shoelace formula turns the area into a logarithm.