SAT Prep Black Book: The Most Effective SAT Strategies Ever Published (30 page)

Example:

In the equilateral triangle
EQI below, all the sides are of equal length, and all the angles are 60 degrees.

I
n an “isosceles” triangle, two of the three sides are the same length as each other, and two of the three angles are the same size as each other.

Exa
mple:

In the isosceles triangle
ISO below, side IS is the same length as side SO. Also,
SIO and  
SOI have the same degree measurement as each other.
 

A “right” triangle is a triangle that includes a ninety-degree angle as one of its three angles.

A special relationship exists between the measurements of the sides of a right triangle: If you take the lengths of the two shorter sides and square them, and then add those two squares together, the resulting amount is the square of the length of the longest side.

Example:

In the right triangle below,
a
2
+
b
2
=
c
2

The expression of this relationship,
a
2
+
b
2
=
c
2
, is called the “Pythagorean Theorem.”

A “Pythagorean triple” is a set of three numbers that can all be the lengths of the sides of the same right triangle. Memorizing four of these sets will make your life easier on the SAT.

Example:

{3, 4, 5}
is a Pythagorean triple because
3
2
+ 4
2
= 5
2
.

{1, 1,
√2} is a Pythagorean triple because
1
2
+ 1
2
=  √2
2

{1,
√3, 2} is a Pythagorean triple because
1
2
+ √3
2
=  2
2

{5, 12, 13}
is a Pythagorean triple because
5
2
+ 12
2
= 13
2

When we multiply each number in a Pythagorean triple by the same number, we get another
Pythagorean triple.

Example:

If we know {3, 4, 5} is a Pythagorean triple, then we also know {6, 8, 10} is a Pythagorean triple, because {6, 8, 10} is what we get when we multiply every number in {3, 4, 5} by 2.

In a {1, 1,
√2} right triangle, the angle measurements are 45
o
, 45
o
, 90
o
.

In a {1,
√3, 2} right triangle, the angle measurements are 30
o
, 60
o
, 90
o
.

Two triangles are “similar triangles” if they have all the same angle measurements.

Between two similar triangles, the relationship between any two corresponding sides is the same as between any other two corresponding sides.

Example:

Triangles
ABC and
DEF below are similar. Side AB has length 8, and side DE has length 24, so every side measurement in
DEF must be three times the corresponding side in
ABC.