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Answer: 35°

Questions 3, 4, 5, and 6 are worded the same, yet they deal with different geometric shapes. It’s important that you learn to deal with angle measures in a variety of geometric figures.

Advanced geometry involves a little more thinking and a whole bunch of tricks.

Some of the nastier questions involve weird geometry—geometry that is hard to figure out at first from the diagrams but which turns out to be really easy if you look hard enough. The main strategy is to separate the big shape into lots of little shapes, and then solve them one by one.

1. What is the area of the shaded thing?

Well, what is that thing? There’s no formula for shapes like this, but don’t panic.

This shape is actually a bunch of circles—the big one (A), two medium ones (B), and the small one (C). You know the formula for the area of a circle (π ×
r
²
), so you can solve this problem by subtracting the two medium circles from the big circle, adding the small circle, and dividing everything by 2. Answer: ½(64 π − 18 π + 4 π) = 25 π. Pretty cool, no?

2. Again, find the shaded area.

The shaded part is what’s left over when you cut quarter circles out of a square. So you can solve for the area of the square
minus
Four quarter circles equal one whole circle, therefore the area of the shaded area equals the area of the square minus the area of a circle with a radius of 2.

Answer: 16
−
4 π

3. Yet again, find the shaded area.

The area of the rectangle is 5 × 4 = 20. The area of the two semicircles = area of one whole circle. Since the whole side of the rectangle = 4, then the radius of the circle is 1, so the area of the circle is π. The triangle can be rearranged into a rectangle whose sides are 2 × 4 = 8.

Answer: 20 − π − 8 = 12 − π

Usually there are some problems that require you to know some coordinate geometry (graphs). We’ll go over the stuff you need to know, but if this doesn’t sound familiar or if you’ve forgotten it, we recommend you get a brand new pad of graph paper and go visit your math teacher at lunchtime. Math teachers love students who come for help eagerly carrying their own graph paper.

This is the basic graph with some points on it:

The
y
-axis goes down-to-up. The
x
-axis goes left-to-right. The origin is the point where they meet.

Any point on a graph has two coordinates. For example, point A on the graph shown has coordinates (2,4), which means that if you start at the origin and want to get to A, you have to go 2 units to the right and 4 units up. For point A, 2 is called its “
x
-coordinate” and 4 is called its “
y
-coordinate.” The origin has coordinates (0,0), and point B has coordinates (
−
1,
−
3).

For any two points, say (
a,b
) and (
c,d
), the distance between them is given by the following formula, which looks fancier than it is:

For example, to get the distance between point A and point B on the graph above, you would plug them into the formula (it doesn’t matter which you call (
a,b
) and (
c,d
)) and get

(If you care why the distance formula works, just draw a right triangle using AB as the hypotenuse, and you’ll see that the formula is just another version of the Pythagorean Theorem.)

Slope

The slope of a line is an indication of how “steep” it is. To figure out the slope of a line that connects two points (
a,b
) and (
c,d
), you use the formula

(Note that the slope of a vertical line is undefined because the change in
x
—the denominator—equals zero.)

JaJa says: The way I remember slope is simply
.