Geometry
Triangles
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:
  1. 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.
  2. The 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.
  3. No side of a triangle can be as large or larger than the sum of the other two sides.
If 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.
Equilateral triangles are a model of uniformity. They have three equal sides and therefore three equal angles, and the interior angles are always 60˚ (603 = 180).
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.
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