NC Math 1: a complete guide to coordinate geometry and reasoning

What this strand demands

This guide covers coordinate geometry and reasoning on the NC Math 1 EOC, drawing on Expressing Geometric Properties with Equations (NC.M1.G-GPE). Geometry is about 8 to 12 percent of the test, the smallest conceptual category, but it appears on every form, so these skills are worth securing. The theme is using algebra on coordinates to settle geometric questions. Each dot-point page has its own practice: slope criteria for parallel and perpendicular lines, coordinate proofs, the distance formula, partitioning a directed segment, and the midpoint formula.

Slope criteria

Two non-vertical lines are parallel when their slopes are equal, and perpendicular when their slopes are negative reciprocals (their product is $-1$). To get a perpendicular slope, flip the fraction and change the sign. A horizontal line (slope $0$) and a vertical line (undefined slope) are perpendicular as a special case.

Coordinate proofs

A coordinate proof uses slope and distance on the vertices. Slope proves sides are parallel (equal slopes) or perpendicular (negative reciprocals); the distance formula proves sides are congruent (equal lengths). So a right triangle has a perpendicular pair of sides, an isosceles triangle has two equal sides, a parallelogram has both pairs of opposite sides parallel, and a rectangle adds a right angle.

The distance formula

The distance between $(x_1, y_1)$ and $(x_2, y_2)$ is $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$, which comes from the Pythagorean theorem (the changes in $x$ and $y$ are the legs, the distance is the hypotenuse). Leave irrational results in exact radical form unless a decimal is asked for.

Partitioning and the midpoint

To partition the segment from $A$ to $B$ in ratio $m:n$ from $A$, move $\frac{m}{m+n}$ of the way: add $\frac{m}{m+n}$ of the total change to $A$. The midpoint is the $1:1$ case, the average of the endpoints: $\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$. To find a missing endpoint, set each average equal to the midpoint coordinate and solve.

How this strand is examined

• Gridded response. Compute a distance, midpoint, or partition coordinate. Exact-match scoring.
• Multiple choice. Identify parallel or perpendicular lines, classify a figure, or choose a slope.
• Technology-enhanced. Match figures to properties, or plot a point.

Check your knowledge

Work these as you would for credit on the EOC.

1. A line has slope $\frac{3}{5}$. What is the slope of a perpendicular line? (1 point)
2. Are $y = 2x + 1$ and $y = 2x - 7$ parallel, perpendicular, or neither? (1 point)
3. Find the distance between $(1, 2)$ and $(4, 6)$. (1 point)
4. Find the exact distance between $(0, 0)$ and $(5, 5)$. (2 points)
5. Find the midpoint of $(2, 3)$ and $(8, 11)$. (1 point)
6. The midpoint of $AB$ is $(4, 4)$ and $A = (2, 1)$. Find $B$. (2 points)
7. Partition from $A(0, 0)$ to $B(8, 12)$ in ratio $1:3$ from $A$. (2 points)
8. A triangle has vertices $(0,0)$, $(4,0)$, $(0,5)$. Is it a right triangle? (1 point)