Miscellaneous Properties

You should be familiar with the properties of parallel lines, and with the rules for deciding when triangles are congruent or similar. In the accompanying figure if $ABC$ is a triangle and $DE$ is parallel to $BC$, and the lengths of the sides are as labeled, you should be able to calculate $DE$.

The Pythagorean Theorem: If $ABC$ is a right triangle with the right angle at $B$, then

$$(AB)^2 + (BC)^2 = (AC)^2.$$ (C.117)

In a given circle, equal arcs subtend equal chords.

A regular hexagon inscribed in a circle has each of its sides equal to the radius of the circle. The radii joining the vertices of this hexagon to the center of the circle decompose the hexagon into six equilateral triangles.

It is assumed that you are familiar with the process of representing points in the plane by pairs of numbers. If $P_1 = (x_1,y_1)$ and $P_2 = (x_2,y_2)$ are points with $x_1 \not= x_2$, then the slope of the segment joining $P_1$ to $P_2$ is defined to be

$$\mbox{slope}(P_1 P_2) =\frac{y_2 - y_1}{x_2 - x_1} = \frac{y_1 - y_2}{x_1 - x_2} = \mbox{slope}(P_2 P_1).$$ (C.118)

If $x_1 = x_2$ we say that $P_1 P_2$ has undefined slope, or that $P_1 P_2$ is a vertical segment. If slope$(P_1 P_2)$ is zero we say that $P_1 P_2$ is a horizontal segment.

Let $P=(p_1,p_2)$ and $Q=(q_1,q_2)$ be two distinct points in the plane. If $p_1 = q_1$, then the line passing through $P$ and $Q$ is defined to be the set of all points of the form $(p_1,y)$, where $y$ can be an arbitrary real number. If $p_1 \not= q_1$, then the line joining $P$ to $Q$ is defined to be the set consisting of $P$ together with all points $X = (x,y)$ such that slope($PX$) = slope($PQ$). Thus if $p_1 \not= q_1$ then $(x,y)$ is on the line joining $P$ to $Q$ if and only if

$$(x,y) = (p_1,p_2) \makebox[5em]{or} \frac{y-p_2}{x-p_1} = \frac{q_2-p_2}{q_1-p_1}.$$ (C.119)

If $m$ = slope($PQ$) then equations C.119 can be rewritten as

$$y = p_2 + m ( x - p_1).$$ (C.120)

If $p_1 \not= q_1$ then (C.120) is called an equation for the line joining $P$ and $Q$. If $p_1 = q_1$ then

$$x = p_1$$ (C.121)

is called an equation for the line joining $P$ and $Q$. Thus a point $X$ is on the line joining $P$ to $Q$ if and only if the coordinates of $X$ satisfy an equation for the line.

Two lines are parallel, (i.e. they do not intersect or they are identical,) if and only if they both have the same slope or they both have undefined slopes.