Up Your Score (41 page)

Also, if you ever see a question that asks you what 1,293,254 to the 0th power is, you can punch it into your calculator, or you can simply remember that
anything (that’s not zero) to the 0th power is equal to 1
. Even if it’s 1,293,254
⁰
. It’s just 1.

Radical equations on the SAT will probably consist of
x
under a radical. When you see something like
, all you have to remember is one thing: The plan of attack is to isolate the enemy. Get the radical by itself. The rest is a piece of cake.

Example: If
, then what is
x
?
The first thing you do in a problem like this is to get all the constants on one side, so you would add 4 to both sides:
Then, in order to get the radical by itself, you have to divide both sides by ½ or multiply both sides by two.

In this case, since
x
was just square-rooted, all we would have to do is square both sides and see that
x
equals 144.

When
y
is
directly proportional
to
x
, that means that an equation
y = kx
can be written (
k
is any constant number). For example,
y
= 5
x
and
y
= (⅕)
x
are directly proportional equations. On the other hand, when
y
is
inversely proportional
to
x
, then the equation is
y
=
k/x
. The equation
y
=
5/x
is inversely proportional.

First off, familiarize yourself with the following symbols, definitions, laws, and formulas.

Symbols:
∥ means “is parallel to.”
l
₁
∥
l
₂
means line 1 is parallel to line 2.

⊥ means “is perpendicular to.”      
l
₁
⊥
l
₂

Congruent angles
have equal numbers of degrees. (They fit perfectly over each other.)

Complementary angles
add up to 90 degrees of arc. (If they look complementary, and it doesn’t say “not drawn to scale,” they probably are complementary. Don’t bother proving it to yourself if you’re pressed for time.)

General Note: Here’s a tip that may help you with geometry problems and possibly with other parts of the math test as well. If the caption for a diagram says “not drawn to scale,” then the first thing you should do is make a quick sketch that
is
to scale. One of the reasons they don’t draw things to scale is to obscure the answer. So, if possible, draw it to scale—nothing elaborate, nothing time-consuming, just a quick sketch. Maybe it will reveal the answer immediately.

Supplementary angles
add up to 180 degrees of arc. (If they look supplementary, they probably are.)

Parallel lines cut by another line:
These things are full of congruent and supplementary angles. You could try to memorize which pairs of angles are congruent and which pairs are supplementary, but why bother? The ones that look supplementary are supplementary, and the ones that look congruent are congruent. In some problems, you may need to extend the lines for it to look like this:

Parallelogram:
Opposite sides are parallel. Parallelograms have two pairs of
equal
(or
congruent
) angles and four pairs of
supplementary
angles. In the diagram, the ones that look equal are equal and the ones that look supplementary are supplementary.

Similar triangles:
Well, boys and girls, now it’s time to give you some exciting insights into similar triangles. But after extensive research, we’ve decided that there is nothing exciting about similar triangles. In fact, there’s been nothing
new
in similar triangles for something over 2,000 years—but (and this is the incredible part) they’re still in fashion with the ETS. So don your toga and get psyched for a bacchanalian triangle party!

What are similar triangles? Similar triangles are two or more triangles with angles of the same measure in different sizes. Here, for example, are two similar triangles:

Continuing along this line of thought, here are two similar fish:

Same shape, different sizes (size 6 and size 10).

The technical way to think about similar triangles is that the two triangles have three angles of corresponding measures. And if you think about it, knowing that
two
of the three angles are the same is enough to ensure that
all three
are the same, since the angles of a triangle always add up to 180°.

So, to jump right in here, what is the measure of angle
x
if A and B are similar triangles?