Use the distance, midpoint, and slope formulas to analyze figures; determine parallel and perpendicular lines from slope; partition a directed segment into a given ratio; and write equations of lines satisfying given conditions.

What this topic is asking

The Regents Geometry exam (the Expressing Geometric Properties with Equations, G-GPE, cluster) wants you to use the distance, midpoint, and slope formulas to analyze figures, decide when lines are parallel or perpendicular from their slopes, partition a directed segment into a given ratio, and write equations of lines meeting stated conditions. These coordinate tools power the coordinate proofs in Parts III and IV.

The three core formulas

These appear constantly, and only the distance formula has a relative on the reference sheet (via the Pythagorean theorem); the slope and midpoint formulas must be known.

Distance tests whether segments are congruent, midpoint tests whether a point bisects a segment or whether diagonals bisect each other, and slope tests parallel and perpendicular.

Parallel and perpendicular lines

Slope decides the relationship between two lines:

• Parallel lines have equal slopes.
• Perpendicular lines have slopes that are negative reciprocals, so their product is $-1$ (for example, $3$ and $-\frac{1}{3}$).

These tests are the engine of coordinate proofs: a right angle is shown by a slope product of $-1$, and parallel sides by equal slopes.

Partitioning a directed segment

To find the point that divides a directed segment from $A$ to $B$ in the ratio $m:n$, move a fraction $\frac{m}{m + n}$ of the way from $A$ toward $B$.

Writing equations of lines

To write the equation of a line, you need a slope and a point. Point-slope form $y - y_1 = m(x - x_1)$ is the most direct, and you can rearrange to slope-intercept form $y = mx + b$. For a line parallel to a given line, reuse its slope; for a perpendicular line, use the negative reciprocal. For instance, the line through $(2, 5)$ perpendicular to a line of slope $4$ has slope $-\frac{1}{4}$, so its equation is $y - 5 = -\frac{1}{4}(x - 2)$, which rearranges to $y = -\frac{1}{4}x + \frac{11}{2}$. A clarifying point that catches many students is the direction of a partition ratio: partitioning from $A$ to $B$ in ratio $1:2$ puts the point one-third of the way from $A$, not two-thirds, so always anchor the fraction $\frac{m}{m + n}$ at the starting point named first. Reversing the direction of the segment changes the answer.

Try this

Q1. Find the midpoint of the segment from $(2, -3)$ to $(8, 5)$. [1 credit]

• Cue. Average the coordinates: $\left(\frac{2 + 8}{2}, \frac{-3 + 5}{2}\right) = (5, 1)$.

Q2. A line has slope $4$. What is the slope of any line perpendicular to it? [1 credit]

• Cue. Negative reciprocal: $-\frac{1}{4}$.