Triangles, triangles, triangles. Get used to the word,
because triangles are the most common figure you will see and use
on SAT geometry items. Triangles are closed figures containing three
angles and three sides.
There are three important rules to know about triangles
on the SAT:
The sum of the three angles in a triangle will always
equal 180˚. You’ll use this fact so many times on the SAT, you may
as well tattoo it to your inner eyelid right now.
longest side of a triangle is always opposite the largest angle; the
second-longest side is always opposite the second-longest angle; and
the shortest side is always opposite the shortest angle.
side of a triangle can be as large or larger than the sum of the other
you know that a triangle has sides of length 4 and 6, you know the third
side is smaller than 10 (since 6 + 4 = 10) and bigger than 2 (since
6- – 4 = 2).
Here are three different types of triangles:
Scalene triangles have no equal sides and
therefore no equal angles. There’s nothing very fiendish or clever
about them, so scalene triangles don’t get much play on the SAT.
Isosceles triangles have two equal sides,
which means they also have two equal angles (the little marks on
two of the sides and angles show that they are congruent,
or equal). Because two of the angles of an isosceles triangle are
equal and because all triangles contain exactly 180˚, if you
know the value of one of the two equal angles, you can figure out
the value of all the angles. Isosceles triangles appear in many
items on the SAT.
triangles are a model of uniformity.
They have three equal sides and therefore three equal angles, and
the interior angles are always 60˚ (603
The formula for the area of a triangle is
You can use any side for the base, but once you pick a base, the
height has to be a line perpendicular to the base that extends up
to the third point of the triangle. On the scalene triangle, you
can see that the height is uneven inside the triangle.
Right Triangles and Pythagoras
A right triangle is a triangle with one 90˚
interior angle (called a right angle). Because the
angles of a triangle must total 180˚, the nonright angles in a right
triangle must add up to 90˚ (180 – 90 = 90). This means that the right
angle is always the largest angle in a right triangle. This also
means the side opposite the right angle, called the hypotenuse,
is the longest side in a right triangle. The Greek mathematician
Pythagoras figured out that if two sides of a right triangle were
known, the value of the third side could also be determined. Of
course you had to use his equation to do it:
a2 + b2 = c2,
where c is the hypotenuse.
Looking at the first triangle, if a =
3 and b = 4, we can determine what c equals.
This right triangle is called a 3:4:5 right triangle after
the length of the sides. Since all three sides are nice round numbers,
the SAT likes this right triangle. It also likes the two special-case
right triangles shown below it, the 30-60-90 right triangle and
the 45-45-90 right triangle.
The 30-60-90 triangle is actually half of an
equilateral triangle. If you imagine an equilateral triangle and
then cut it down the middle, you’ll end up with a 30-60-90. As the
second figure shows, the ratio between the three sides is always
the same. The side opposite the 90˚ angle is always twice as long
as the side opposite the 30˚ angle. The side opposite to the 60˚
angle is always
times as long as
the side opposite the 30˚ angle.
Learn these two special-case right triangles by heart.
Tattoo them to your inner eyelid if you have any extra space.