SL 3.5—Intersection of lines, equations of perpendicular bisectors

The midpoint of $(x_1,y_1)$ and $(x_2,y_2)$ is $M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$.

With $A(-3,5)$ and $B(9,-1)$: $M=\left(\frac{-3+9}{2},\frac{5+(-1)}{2}\right)$.

$M=\left(\frac{6}{2},\frac{4}{2}\right)=(3,2)$.

The midpoint is $(3,2)$.

$(6,4)$ comes from summing coordinates without dividing by 2.

$(3,-2)$ results from a sign error when averaging the $y$-coordinates: $5+(-1)=4$, not $-4$.

$(-6,-4)$ uses the differences of coordinates rather than the sums, and forgets to halve.

Only $(3,2)$ correctly averages both coordinates.

The perpendicular slope is the negative reciprocal: $m_{\perp }=-\frac{1}{m}$.

With $m=-\frac{4}{5}$: $m_{\perp }=-\frac{1}{-\frac{4}{5}}$.

$m_{\perp }=\frac{5}{4}$ since flipping and negating a negative fraction gives a positive result.

The perpendicular bisector has slope $\frac{5}{4}$.

$-\frac{5}{4}$ flips the fraction but incorrectly keeps the negative sign.

$\frac{4}{5}$ forgets to flip the fraction, only removing the negative sign.

$-\frac{4}{5}$ is simply the original slope unchanged, with no reciprocal or sign flip applied.

Only $\frac{5}{4}$ correctly flips and negates the original slope.