-An isosceles right triangle (45-45-90) has sides in a ratio of x : x : x√2
-An equilateral triangle has three equal side. Each angle is equal to 60 degrees
-Any given angle of a triangle corresponds to the length of the opposite side. The larger the degree measure of the angle, the larger the length of the opposite side.
-Each side of certain right triangles are integers (e.g., 3 : 4 : 5, 5 : 12 : 13).
-The length of the longest side can never be greater than the sum of the two other sides.
-The length of the shortest side can never be less than the positive difference of the other two sides.
Circumference for a circle.
or (r=radius & D=diameter)
Area of a circle.
-A fraction of the circumference of a circle is called an arc. To find the degree measure of an arc, look at the central angle.
-Now the properties of inscribed squares (If x is the side of the square the diameter of the circle will equal x√2).
-The area of a square is s2 (s = side).
-The diagonals of a square bisect one another, forming four 90 degree angles
-The diagonals of a rhombus bisect on another, forming four 90 degree angles
-Twice the length plus the width equals the perimeter of a rectangle
-The area of a parallelogram can be found multiplying base x height (the base always forms a right angle with the height).
SUrface Area of a cube.
Volume of a Cube
The volume of a cube and the surface area of a cube are equal when s = 6.
Volume of a Cylinder
2h (h=height & r=radius)
Volume of a Sphere
-The slope of a line can be found subtracting the y values of a pair of coordinates and dividing it by the difference in the x values
-To find the y-intercept plug in zero for x
-To find the x-intercept, plug in zero for y and solve for x
-The slopes of two lines which are perpendicular to each other are in the ratio of x : -1/x, where x is the slope of one of the lines (think: negative reciprocal).
Odds & Evens
even +/- even = even
even +/- odd = odd
odd +/- odd = even
even × even = even
even × odd = even
odd × odd = odd
Divisibility Rule for #2
A number is divisible by 2 if it’s even.
Divisibility Rule for #3.
A number is divisible by 3 if the sum of its digits is divisible by 3. This means you add up all the digits of the original number. If that total is divisible by 3, then so is the number. For example, to see whether 83,503 is divisible by 3, we calculate 8 + 3 + 5 + 0 + 3 = 19. 19 is not divisible by 3, so neither is 83,503.
Divisibility for the #4.
A number is divisible by 4 if its last two digits, taken as a single number, are divisible by 4. For example, 179,316 is divisible by 4 because 16 is divisible by 4.
Divisibility for #5.
A number is divisible by 5 if its last digit is 0 or 5. Examples include 0, 430, and –20.
Divisibility for #6.
A number is divisible by 6 if it’s divisible by both 2 and 3. For example, 663 is not divisible by 6 because it’s not divisible by 2. But 570 is divisible by 6 because it’s divisible by both 2 and 3 (5 + 7 + 0 = 12, and 12 is divisible by 3).
Divisibility for #7.
7 may be a lucky number in general, but it’s unlucky when it comes to divisibility. Although a divisibility rule for 7 does exist, it’s much harder than dividing the original number by 7 and seeing if the result is an integer. So if the GRE happens to throw a “divisible by 7” question at you, you’ll just have to suck it up and do the math.
Divisibility Rule for #8.
A number is divisible by 8 if its last three digits, taken as a single number, are divisible by 8. For example, 179,128 is divisible by 8 because 128 is divisible by 8.
Divisibility Rule for #9.
A number is divisible by 9 if the sum of its digits is divisible by 9. This means you add up all the digits of the original number. If that total is divisible by 9, then so is the number. For example, to see whether 531 is divisible by 9, we calculate 5 + 3 + 1 = 9. Since 9 is divisible by 9, 531 is as well.
Divisibility Rule for #10.
A number is divisible by 10 if the units digit is a 0. For example, 0, 490, and –20 are all divisible by 10.
Divisibility Rule for #11.
Add every other digit starting with the leftmost digit and write their sum. Then add all the numbers that you didn’t add in the first step and write their sum. If the difference between the two sums is divisible by 11, then so is the original number. For example, to test whether 803,715 is divisible by 11, we first add 8 + 3 + 1 = 12. To do this, we just started with the leftmost digit and added alternating digits. Now we add the numbers that we didn’t add in the first step: 0 + 7 + 5 = 12. Finally, we take the difference between these two sums: 12 – 12 = 0. Zero is divisible by all numbers, including 11, so 803,715 is divisible by 11.
Divisibility Rule for #12.
A number is divisible by 12 if it’s divisible by both 3 and 4. For example, 663 is not divisible by 12 because it’s not divisible by 4. 162,480 is divisible by 12 because it’s divisible by both 4 (the last two digits, 80, are divisible by 4) and 3 (1 + 6 + 2 + 4 + 8 + 0 = 21, and 21 is divisible by 3).
Keep things simple, and use the Magic X. The key is to multiply diagonally and up, which in this case means from the 9 to the 3 and also from the 7 to the 2:
In an addition problem, we add the products to get our numerator: 27 + 14 = 41(Subtract for subtraction). For the denominator, we simply multiply the two denominators to get 63. So the answer is 41/63.
The Magic X: Quantitative Comparison.
Compare which fraction is larger: 5/23 or 7/30?Again, begin by multiplying diagonally and up:
Now compare the numbers you get: 161 is larger than 150, so 7/30 is greater than 5/23.
Percent increase/decrese formula
Product & Quotient Rule (multiplying and dividing exponents):