The distance between points and the midpoint

Analytic geometry studies shapes by numbers: a point is a pair of numbers $\displaystyle (x,y)$, and a line or a circle is an equation that the coordinates of its points satisfy.

• René Descartes and Pierre de Fermat, in the 17th century, joined algebra and geometry this way – the coordinate system is named “Cartesian” after Descartes.
• A question about a shape becomes an equation: where two lines meet is the solution of a system of equations.
• An equation becomes a picture: $\displaystyle {y}={x^{2}}$ is a parabola you can see.

The distance between two points is the hypotenuse of a right-angled triangle whose legs are parallel to the axes, so Pythagoras’ theorem gives it.

• The leg $\displaystyle AC$ is parallel to the $\displaystyle x$-axis: its length is the difference between the $\displaystyle x$-coordinates, $\displaystyle {7-1}={6}$.
• The leg $\displaystyle BC$ is parallel to the $\displaystyle y$-axis: its length is the difference between the $\displaystyle y$-coordinates, $\displaystyle {5-1}={4}$.
• By Pythagoras’ theorem: $\displaystyle {AB}={\sqrt{6^{2}+4^{2}}}={\sqrt{52}}={2\sqrt{13}}$.

For any two points $\displaystyle A(x_1,y_1)$ and $\displaystyle B(x_2,y_2)$:

• The order of the points makes no difference: each difference is squared.
• A distance you go on working with stays a root in its simplest form: $\displaystyle 2\sqrt{13}$, not $\displaystyle 7.21$.

Find the distances.

Each coordinate of the midpoint is the mean of the coordinates of the ends:

Why: $\displaystyle M$ is the midpoint of $\displaystyle AB$, and $\displaystyle AK$, $\displaystyle ML$ are parallel to the $\displaystyle x$-axis.

• The triangles $\displaystyle AKM$ and $\displaystyle MLB$ are congruent (ASA): $\displaystyle {AM}={MB}$; the angles at $\displaystyle A$ and at $\displaystyle M$ are corresponding angles between parallel lines; the angles at $\displaystyle K$ and at $\displaystyle L$ are right.
• So $\displaystyle {AK}={ML}$: the step in $\displaystyle x$ from $\displaystyle A$ to $\displaystyle M$ equals the step from $\displaystyle M$ to $\displaystyle B$, $\displaystyle {x_M-x_1}={x_2-x_M}$, and $\displaystyle {x_M}={\frac{x_1+x_2}{2}}$.
• In the same way $\displaystyle {KM}={LB}$ gives $\displaystyle {y_M}={\frac{y_1+y_2}{2}}$.

Example: $\displaystyle A(1,1)$ and $\displaystyle B(7,5)$: $\displaystyle {M\left(\frac{1+7}{2},\frac{1+5}{2}\right)}={(4,3)}$.

Find the midpoint of the segment $\displaystyle PQ$: $\displaystyle P(-3,8)$ and $\displaystyle Q(5,2)$.

$\displaystyle M(2,-1)$ is the midpoint of the segment $\displaystyle AB$, and $\displaystyle A(-4,3)$.
Find the coordinates of $\displaystyle B$.

A point at equal distances from $\displaystyle A$ and from $\displaystyle B$ gives an equation: the two distances are equal, and so are their squares.

Example: the point $\displaystyle P$ on the $\displaystyle y$-axis at equal distances from $\displaystyle A(2,1)$ and $\displaystyle B(6,5)$.

• $\displaystyle P$ is on the $\displaystyle y$-axis, so $\displaystyle P(0,y)$.
• $\displaystyle {PA^{2}}={PB^{2}}$: $\displaystyle {(0-2)^{2}+(y-1)^{2}}={(0-6)^{2}+(y-5)^{2}}$.
• $\displaystyle {4+y^{2}-2y+1}={36+y^{2}-10y+25}$, so $\displaystyle {8y}={56}$ and $\displaystyle {y}={7}$.
• $\displaystyle P(0,7)$. A check: $\displaystyle {PA^{2}}={4+36}={40}$ and $\displaystyle {PB^{2}}={36+4}={40}$.

The $\displaystyle y^{2}$ cancels: such an equation is always of the first degree.

The point $\displaystyle P$ is on the $\displaystyle x$-axis, at equal distances from $\displaystyle A(1,3)$ and $\displaystyle B(5,1)$.
Find $\displaystyle P$.

The vertices of the triangle $\displaystyle ABC$ are $\displaystyle A(-1,1)$, $\displaystyle B(7,1)$ and $\displaystyle C(3,4)$.

Find the lengths of the sides $\displaystyle AC$ and $\displaystyle BC$.

$\displaystyle D$ is the midpoint of $\displaystyle AB$.
Find the length of the median $\displaystyle CD$.

What kind of triangle is $\displaystyle ABC$?
Explain why the median $\displaystyle CD$ is also the altitude to $\displaystyle AB$.