From the complexities of closed figures,
let's simplify things and think only about lines extending in
a plane. The two most important facts about arbitrary lines
are that (a) when they cross the opposite angles are equal and
(b) on one side of a line, there is always 180 degrees. [In
fact, you can derive (a) by knowing (b).]
Because there are
180o in a straight <, we know
x + z = 180o
But we also know that
y + z = 180o
Therefore x must equal y.
Parallel lines
When two lines run so that they never intersect (even if projected
an infinite distance in either direction), they are called parallel
lines. When two parallel lines are crossed by a third, there
is a special correspondence in the angles that are formed.
The
marked angles in the figure above are equal.
Okay, let's put it all together. What
can you say about the two blue angles marked in this picture?
If it doesn't come to you right away, review the previous
few pages and look for the hints that you need.
Answer:
The angles are equal. How to see this? Try using: (1) the
fact that the sum of the angles on one side of a line is 180
degrees and (2) the equality of the two angles from the previous
page.
Coordinate Geometry
Sometimes in a big city roads going in one direction are called
avenues, and the roads crossing these are called streets. The
avenues and streets are named with consecutive numbers. That
way, to identify any spot in the city, you just need to give
the intersection between an avenue and a street.
"Meet
me at 56th Street and 7th Avenue" works because the corner
of 56th and 7th is a single point in the city. Furthermore,
when you know two points, you also know the distance between
them.
"Let's walk from 56th and 7th to 83rd and 7th."
"Oh man! That's 27 blocks!"
How do you know? Because 83 blocks - 56 blocks = 27 blocks.
In coordinate geometry we do exactly the
same thing. (Okay, let's be clear, it's the city planners who
borrowed from coordinate geometry.) Here's how they are similar.
Label the vertical axis the y-axis. Label the horizontal
axis the x-axis. Instead of blocks we'll talk about units.
Just as in the city, any point in the coordinate plane can
be identified by a pair of x, y coordinates, usually presented
as a pair like this: (x,y).
The point (3,2) is a unique point in the grid, at the location
x = 3 and y = 2.
The x-axis and the y-axis intersect at the origin, the point
with coordinates (0,0).
With coordinate geometry we can do some
interesting things. For instance, it is easy to find the distance
between two points.
Take the points (3,2) and (7,5). To go from (3,2) to (7,5)
you need to move 4 units along the x axis, and 3 units up
the y-axis.
So how far is that? Well, in terms of city blocks, it would
be 4 + 3 = 7 blocks.
But we can also use the Pythagorean theorem to get the shortest
distance, which is
42 + 32
= 22 = 52
So the straight line from point (3,2) to point (7,5) is 5
units long.
Slopes
Between any two points, one unique straight line can be drawn.
The slope of that line tells us about its orientation to the
grid. Lines that drop downward, from left to right across the
grid have a negative slope; lines that are parallel to the x-axis
have zero slope; and lines that rise as they move from left
to right have positive slope.
A very good way to think about slope is as the rise over
run, or equivalently, the change in y divided by the change
in x.
For any two points, the change in y, as you move from one
point to the next, divided by the change in x, is the slope.
For example, the slope of the line between
points (3,2) and (7,5) is
m = (5 - 2) / (7 - 3) = 3 / 4
The slope is positive (as we can see from the picture - it
rise from left to right), and has a value of .75.
What does slope mean? Slopes can be very informative, especially
when we know the units of the two axes. For instance, suppose
the y-axis represented distance and the x-axis represented
time. Then the slope, y/x, represents the change in distance
for a corresponding change in time. That is a quantity we
also know as speed - measured in, for example, miles/hour.
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