Use the distance formula, the midpoint formula, and slope to find lengths and midpoints, classify figures, and verify properties such as parallel and perpendicular sides on the coordinate plane.

What this topic is asking

The Geometry category includes coordinate geometry (the G-GPE standards): using algebra on the coordinate plane to find lengths and midpoints and to classify figures. On the Grade 10 MCAS you apply the distance and midpoint formulas and use slope to test for parallel and perpendicular sides. The distance formula is not on the reference sheet (it is the Pythagorean theorem in disguise), so you must know it, along with the midpoint formula.

The distance formula

The distance between two points is the length of the segment joining them. It comes straight from the Pythagorean theorem: the horizontal gap and the vertical gap are the legs, and the distance is the hypotenuse.

For $(1, 2)$ and $(4, 6)$: $d = \sqrt{(4 - 1)^2 + (6 - 2)^2} = \sqrt{9 + 16} = \sqrt{25} = 5$. Square each difference, add, then take the root; skipping the root or the squaring are the usual errors. Because it is not on the reference sheet, this formula must be memorized, though you can always reconstruct it from $a^2 + b^2 = c^2$.

The midpoint formula

The midpoint of a segment is the point exactly halfway between the endpoints, found by averaging the coordinates:

For $(1, 2)$ and $(4, 6)$: $M = \left(\frac{1 + 4}{2}, \frac{2 + 6}{2}\right) = \left(\frac{5}{2}, 4\right)$. Averaging, not subtracting, is the key; the midpoint is a balance point of the two endpoints.

Slope for parallel and perpendicular

Slope on the coordinate plane tests how two segments relate:

• Parallel segments have equal slopes.
• Perpendicular segments have slopes that are negative reciprocals, so their product is $-1$. A slope of $\frac{2}{3}$ is perpendicular to $-\frac{3}{2}$.

These tests let you verify a figure's properties. To show a quadrilateral is a parallelogram, show both pairs of opposite sides have equal slopes; to show a triangle has a right angle, show two sides have perpendicular slopes.

Putting the tools together

A typical MCAS coordinate task combines the three tools: distance for side lengths, midpoint for centers or diagonals, and slope for parallel or perpendicular sides. For example, to show a quadrilateral is a rectangle, you might check that opposite sides are parallel (equal slopes) and that adjacent sides are perpendicular (slopes multiply to $-1$). To check a parallelogram's diagonals bisect each other, show they share the same midpoint.

Try this

Q1. Find the midpoint of $(-2, 5)$ and $(4, -1)$.

• Cue. $\left(\frac{-2 + 4}{2}, \frac{5 + (-1)}{2}\right) = (1, 2)$.

Q2. Are lines with slopes $\frac{3}{4}$ and $-\frac{4}{3}$ parallel or perpendicular?

• Cue. Product is $-1$, so perpendicular.