Multiplication Table 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
2 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
3 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60
Square Roots √2 = 1.41 √3 = 1.73 √5 = 2.236 √6 = 2.449 √7 = 2.646 √10 =3.16
4 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80
5 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
6 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90 96 102 108 114 120
7 7 14 21 28 35 42 49 56 63 70 77 84 91 98 105 112 119 126 133 140
8 8 16 24 32 40 48 56 64 72 80 88 96 104 112 120 128 136 144 152 160
9 9 18 27 36 45 54 63 72 81 90 99 108 117 126 135 144 153 162 171 180
10 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200
11 11 22 33 44 55 66 77 88 99 110 121 132 143 154 165 176 187 198 209 220
12 12 24 36 48 60 72 84 96 108 120 132 144 156 168 180 192 204 216 228 240
13 13 26 39 52 65 78 91 104 117 130 143 156 169 182 195 208 221 234 247 260
14 14 28 42 56 70 84 98 112 126 140 154 168 182 196 210 224 238 252 266 280
15 15 30 45 60 75 90 105 120 135 150 165 180 195 210 225 240 255 270 285 300
16 16 32 48 64 80 96 112 128 144 160 176 192 208 224 240 256 272 288 304 320
17 17 34 51 68 85 102 119 136 153 170 187 204 221 238 255 272 289 306 323 340
18 18 36 54 72 90 108 126 144 162 180 198 216 234 252 270 288 306 324 342 360
19 19 38 57 76 95 114 133 152 171 190 209 228 247 266 285 304 323 342 361 380
20 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400
Squares and Cubes Number ( x ) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 21 22 23 24 25
Square ( x 2 ) 1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 256 289 324 361 441 484 529 576 625
Cube ( x 3 ) 1 8 27 64 125 216 -
Fractions and Percentage Fraction 1/2 1/3 2/3 1/4 3/4 1/5 2/5 3/5 4/5 1/6 5/6 1/8 3/8 5/8 7/8 1/9 2/9 1 / 10 1 / 20 1 / 100
Decimal 0.5 0.33 0.66 0.25 0.75 0.2 0.4 0.6 0.8 0.166 0.833 0.125 0.375 0.625 0.875 0.111 0.222 0.1 0.05 0.01
Percentage 50 33 1/3 66 2/3 25 75 20 40 60 80 16 2/3 83 1/3 12 1/2 37 1/2 62 1/2 87 1/2 11 22 10 5 1
Natural numbers are 1,2,3… Whole numbers are 0,1,2,3 Integers are 0,±1, ±2, ±3,… Composite number is opposite of prime number Rational number can be expressed as p/q Irrational number cannot be expressed as p/q such as √2 A fraction will result in a terminating decimal only if the denominator is written in form of prime numbers of 2 and 5. Otherwise the fraction will not result in a terminating decimal.
Prime Numbers 0 and 1 are not prime numbers. Here is a table of all prime numbers up to 1,000: 2
3
5
7
11 13 17 19 23
29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281 283 293 307 311 313 317 331 337 347 349 353 359 367 373 379 383 389 397 401 409 419 421 431 433 439 443 449 457 461 463 467 479 487 491 499 503 509 521 523 541 547 557 563 569 571 577 587 593 599 601 607 613 617 619 631 641 643 647 653 659 661 673 677 683 691 701 709 719 727 733 739 743 751 757 761 769 773 787 797 809 811 821 823 827 829 839 853 857 859 863 877 881 883 887 907 911 919 929 937 941 947 953 967 971 977 983 991 997
Method to determine a prime number: Find approx square root of the number. Then check if all the prime numbers below the square root are factors of the given number. If none are then the number is prime else not. e.g. number 91. approx sq root is 10 Prime number below 10 are 2.3.5&7. 91 is not divisible by2,3 or 5. But it is divisible by 7. Therefore 91 is not prime.
Divisibility Divisibility by 2, 4, 8, 16, 32.. A number is divisible by 2, 4, 8, 16, 32,…2 n when the number formed by the last one, two, three, four, five...n digits is divisible by 2, 4, 8, 16, 32,..2 n respectively. i.e 1246384 is divisible by 8 because the # formed by the last 3 digits i.e. 384 is divisible by 8.
Divisibility by 3 and 9 A number is divisible by 3 or 9 when the sum of the digits of the number is divisible by 3 or 9 respectively. i.e. 3144 is divisible by 3 because the sum of the digits- 3 + 1 + 4 + 4 = 12 is divisible by 3. i.e. 8406 is divisible by 9 because the sum of the digit- 8 + 4 + 0 + 6 = 18 is divisible by 9.
Divisibility by 5, 10 Divisible by 5 if the number ends in 0 or 5 Divisible by 10 if the number ends in 0
Divisibility by 11 11 - Start with the units digit, add every other digit and remember this number. Form a new number by adding the digits that remain. If the difference between these two numbers is divisible by 11, then the original number is divisible by 11. eg. Is the number 824472 divisible by 11? Starting with the units digit, add every other number:2 + 4 + 2 = 8. Then add the remaining numbers: 7 + 4 + 8 = 19. Since the difference between these two sums is 11, which is divisible by 11, 824472 is divisible by 11. When any number with even number of digits is added to its reverse, the sum is always divisible by 11. e.q: 1234+4321 is divisible by 11
Divisibility by 7, 11, and 13 Let a number be ....kjihgfedcba where a, b, c, d, are respectively units digits, tens digits, hundreds digits, thousands digits and so on. Starting from right to left, we make groups of three digit numbers successively and continue till the end. It is not necessary that the leftmost group has three digits. Grouping of the above number in groups of three, from right to left, is done in the following manner kj,ihg,fed,cba We add the alternate groups (1 st , 3 rd , 5 th etc.. and 2 nd , 4 th , 6 th , etc..) to obtain two sets of numbers, N 1 and N 2 . In the above example, N 1 = cba + ihg and N 2 = fed + kj Let D be difference of two numbers, N 1 and N 2 i.e. D = N 1 - N 2 . - If D is divisible by 7, then the original number is divisible by 7. - If D is divisible by 11, then the original number is divisible by 11 - If D is divisible by 13 then the original number is divisible by 13.
Corollary: Any six-digit, or twelve-digit, or eighteen-digit, or any such number with number of digits equal to multiple of 6, is divisible by EACH of 7, 11 and 13 if all of its digits are same . For example 666666, 888888888888 etc. are all divisible by 7, 11, and 13. Example Find if the number 29088276 is divisible by 7. Answer: We make the groups of three as said above- 29,088,276 N 1 = 29 + 276 = 305 and N 2 = 88 D = N 1 - N 2 = 305-88 = 217. We can see that D is divisible by 7. Hence, the original number is divisible by 7.
Divisibility by 25 and 125 A number is divisible by 25 and 125 when the number formed by the last two and three right hand digits are divisible by 25 and 125 respectively. 1025, 3475 divisible by 25 2125, 5375 divisible by 125
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Interest Rates (
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P = principal amount (initial investment) r = annual nominal interest rate (as a decimal) n = number of times the interest is compounded per year t = number of years A = amount after time t
Note: If the interest in compounded annually but time is in fraction i.e 3.4 years, (
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Example: An amount of $1500.00 is deposited in a bank paying an annual interest rate of 4.3%, compounded quarterly. Find the balance after 6 years. (
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When the rates are different for different years A= P(1+R1/100)(1+R2/100)(1+R3/100) Note: No need to calculate the compound interest rate, just calculate the simple interest and the answer will be slightly higher.
Descriptive Statistics The average or (arithmetic) mean of n numbers is defined as the sum of the n numbers divided by n. ∑ . For example, the average of 6, 4, 7, 10, and 4 is (6+4+7+10)/4 = 6.2 The median is another type of center for a list of numbers. To calculate the median of n numbers, first order the numbers from least to greatest; if n is odd, the median is defined as the middle number, whereas if n is even, the median is defined as the average of the two middle numbers. In the example above, the numbers, in order, are 4, 4, 6, 7, 10, and the median is 6, the middle number. For the numbers 4, 6, 6, 8, 9, 12, the median is (6 + 8)/2. The mode of a list of numbers is the number that occurs most frequently in the list. For example, the mode of 1, 3, 6, 4, 3, 5 is 3. A list of numbers may have more than one mode. For example, the list 1, 2, 3, 3, 3, 5, 7, 10, 10, 10, 20 has two modes, 3 and 10. The degree to which numerical data are spread out or dispersed can be measured in many ways. The simplest measure of dispersion is the range, which is defined as the greatest value in the numerical data minus the least value. For example, the range of 11, 10, 5, 13, 21 is 21 – 5 = 16. Note how the range depends on only two values in the data.
Standard Deviation 1. |Median-Mean| <= SD. Standard deviation of a set of numbers can never be larger than the range of the set. 2. Variance is the square of the standard deviation. 3. If Range or SD of a list is 0, then the list will contain all identical elements. And vise versa: if a list contains all identical elements then the range and SD of a list is 0. If the list contains 1 element: Range is zero and SD is zero. 4. SD is always >=0. SD is 0 only when the list contains all identical elements (or only 1 element). 6. If we add or subtract a constant to each term in a set: Mean will increase or decrease by the same constant. SD will not change. 7. If we increase or decrease each term in a set by the same percent: Mean will increase or decrease by the same percent.SD will increase or decrease by the same percent. 8. Changing the signs of the element of a set (multiplying by -1) has no effect on SD. 9. The SD of any list is not dependent on the average, but on the deviation of the numbers from the average. So just by knowing that two lists having different averages doesn't say anything about their standard deviation - different averages can have the same SD.
General 1. In order for x and y to be consecutive perfect squares, given that x is greater than y, it would have to be true that: √ √ 2. Remember to include '1' if you're asked to count the number of factors a number has 3. Not as tall as means may be shorter or taller. i.e. exactly not the same height as 4. If it is difficult to find solve inequality with modes then square the inequality ... this makes things easier. 5. All squares of even numbers must be multiples of 4 6. If a number is of 2 digit then consider 10x+y kind of equation and so on 7. Sum of integers that are formed by the permutations of n digits is given by equation: = (sum of digits)*(n-1)!*(111... n times) if repetition is not allowed. = (sum of digits)*(n)(n-1)*(111... n times)
if repetition is allowed.
i.e What is the sum of all 3 digit positive integers that can be formed using the digits 1, 5, and 8, if the digits are allowed to repeat within a number? A. 126 B. 1386 C. 3108 D. 308 E. 13986 Here n = 3, sum of digits = 14 thus for 3 digits we have to take 111 only. So sum = 14*3^2*(111)=13986 if repetition is not allowed, then sum = 14 *2!*111 =3108 8. If a+b+c = constant then a2b3c4 have maximum values when a, b and c are in ratio 2:3:4 e.g volume of cylinder V = , and given that r+h=9 then maximum value of V will be when r and h are in ratio 2:1 9. What is remainder when 1421*1423*1425 is divided by 12 classical way to solve this is to multiply all the numbers and then do the division operation to get remainder. however, we can get remainder following way: divide each number with given number separately go get the reminder of each number then multiply, do the operation as many times till resultant number is no more divisible by given number... it's little bit abstract: Find the remainder of 1421*1423*1425 when divided by 12 Rof( 1421*1423*1425) /12 ---------> Rof(5*7*9)/12 = Rof(35*9)/12 Rof(11*9)/12 ---> gives us reminder of 3.
Some math solving strategies: 1. Think for 5 sec before starting calculation 2. Picking numbers is almost always the best to attack even/ odd questions. 3. Generally answers are arranged in ascending order, if picking number is the strategy then checking answer choice C, so it will be clear whether we need to check D&E or A&B 5. First 10 +ve multiple of 5 are: 5,10,15,20,25,30,35,40..........( this is example to show what is multiple of some number means ) 7. e.g. if -20 then -b <= x <= b...that is x lies between -b & b, inclusive II. if |x| >= b & b>0 , then either x <= -b or x >= b...that is x lies outside the range -b to b, exclusive. 10. As soon as we see one number can be either +ve or -ve we know the statement is not sufficient. e.g x2 > 0; since x can be +ve or -ve so not possible to determine. 11. Is x multiple of y => is x/y an integer 12. Some basic trigonometric function and values. Recognize that the sine and cosine functions all have the form √n/2 as follows:
In order to reproduce the above table, you also need to remember that sin increases from 0 to 1 in Quadrant I, while cos decreases from 1 to 0 in Quadrant I.
LCM & HCF Lowest Common Multiple Highest Common Factor Example: 12 80 50 12 = 2 2 3 80 = 2 5 2 2 2 50 = 2 5 5 HCF = 2 LCM = 2 * 2 * 2 * 2 * 3* 5 * 5 LCM of Two Numbers * HCF of Two Numbers = Product of 2 Numbers.
Average Speed ( (
Equal Distance, Different Speed, Average speed = Equal Time, Different Distance Average Speed =
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Roots
Basic Factorization Manipulation Formula
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Series Sum of first n natural numbers: 1+2+3+4+...+n = n*(n+1)/2 Sum of square of first n natural numbers: 12+22+32+...+n2 = n*(n+1)(2n+1)/6 Sum of cube of first n natural numbers: 13+23+33+…+n3 = [n*(n+1)/2]2 Sum of first odd natural numbers: 1+3+5+...+ (2k+1) = [n]2
where n=(2k+1)
Sum of first even natural numbers: 2+4+6+...+ (2k) = n(n+1)
where n=(2k)
Square roots √(
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Roots of Quadratic Equation
√
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Ratios Basically if
Basically, you can do all kinds of additions and subtractions. Example:
What is a? From (1) we get So
Absolute values The way to solve this kind of questions is to break the equation (inequality) into two parts, one is when the value is non negative, the other is when the value is negative. For example: |x-4|<9 You break it into two parts: If x-4>=0, then x-4<9, solve for both you get x>=4, x<13. So your solution is 4<=x<13. If x-4<0, then -(x-4)<9, ie x-4>-9. Solve for both you get x<4, x>-5. So your solution for this part is -5|y+1| if y>=0, y+1>=0, y>y+1, no solution. if y<0, y+1<0, -y>-(y+1), solution is y<-1 if y>=0, y+1<0, y>-(y+1), no solution. if y<0, y+1>=0, -y>y+1, solution is -1<=y<-1/2 So your final solution is y<-1/2 You could also solve this question by squaring the terms since both sides are positive. y^2>(y+1)^2 y^2>y^2+2y+1 2y+1<0 y<-1/2
If d is POSITIVE and |x| < d, then -d < x < d If d is NEGATIVE and |x| < d, then there is no solution If d is POSITIVE and |x| > d, then x < -d OR x > d If d is NEGATIVE and |x| > d, then x is all real numbers
Cancelling out "Common Terms" on Both Sides of an Equation You need to be very careful when you do algebra derivations. One of the common mistakes is to divide both side by "a common term
Example: x(x-2)=x You can't cancel out the x on both side and say x=3 is the solution. You must move the x on the right side to the left side. x(x-2)-x=0 x(x-2-1)=0 The solutions are: x=0 and x=3 The reason why you can't divided both sides by x is that when x is zero, you can't divide anything by zero. Equally important if not more, is that you CAN'T multiple or divide a "common term" that includes a variable from both side of an inequality. Not only it could be zero, but it could also be negative in which case you would need to flip the sign. Example: x^2>x You CAN'T divided both sides by x and say x>1. What you have to do is to move the right side to the left: x^2-x>0 x(x-1)>0 Solution would be either both x and x-1 are greater than zero, or both x and x-1 are smaller than zero. So your solution is: x>1 or x<0 Example: x>1/x Again you CAN'T multiply both sides by x because you don't know if x is positive or negative. What you have to do is to move the right side to the left: x-1/x>0 (x^2-1)/x>0 If x>0 then x^2-1>0 =>x>1 If x<0 then x^2-1<0 =>x>-1 Therefore your solution is x>1 or 0>x>-1. You could also break the original question to two branches from the beginning: x>1/x if x>0 then x^2>1 =>x>1 if x<0 then x^2<1 => x>-1 Therefore your solution is x>1 or 0>x>-1.
Basic Rules for Inequalities:
Transitive Property: If x < y and y < z then x < z For any a & b, you need to flip signs when both sides are multiplied by a negative number:
You can only add or multiply them when their signs are in the same direction: You can only apply subtractions and divisions when their signs are in the opposite direction:
You can only add them when their signs are in the same direction: You can only multiply them when their signs are in the same direction but you have to flip the sign because you are multiplying by a negative number: You can only apply subtractions when their signs are in the opposite direction: You can only apply divisions when their signs are in the opposite direction but you have to flip the sign because you are multiplying by a negative number:
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For the reciprocal of the number, you need to flip signs if both are positive or both are negative:
For the reciprocal of the number, if one number is positive and the second is negative do not flip signs.
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Solution of Inequalities: (x-a)*(x-b) > 0
(x-a)*(x-b) < 0
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Work and Rate Problem (
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\: Example: Sam and Roy finish a job in 5 and 4 hours respectively. In how many hours can they finish the job if they work together? Their equivalent rate is
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Example: It takes 6 printers 4 hours to print 5000 papers. How long will it take 3 printers to print 3000 papers? (
It will take 24/5 hours.
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Miscellaneous Rules -
Cannot multiply / divide by variables unless you know their signs. If x>y, this does not necessarily mean that x2 >y2 or that √x > √y. Even powers cannot be predicted.
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If x>y, then x >y and √
3
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√ . Odd powers are predictable.
- |x+y| ≤ |x| +|y| -
0 is multiple of all non-zero integers but is not a factor of any integer. If an expression F(x) is divided by (x-a), then the remainder is F(a). I.e The remainder of F(x)= x3 +3 x2-5 x+4 when divided by (x-1) is F(1) =1+3-5+4=3
Arithmetic Progression
If n is odd, Middle Term = (a + Tn)/2 Sn = Middle Term * n Number of odd numbers is (n+1)/2 Number of even numbers is (n-1)/2 Example 1,2,3; There are (3+1)/2 odd numbers which are 1,3 & (3-1)/2 even number which is 2 Geometric Progression
Consecutive Numbers
Sum of a Series A quick way to find the sum of a series where each preceding term is incremented by the same number would be to find the middle term and multiply it by the number of terms. The middle term can be found at taking the average of the first and last term if there are odd terms. E.g. Sum of 4,8,12,16,20 Middle term: 12 Number of terms: 5 Sum = 12*5=60
For consecutive integers or equally spaced numbers the average = (First + Last)/2 Miscellaneous Notes -
Product of any 3 consecutive numbers is divisible by 6. Product of any 4 consecutive numbers is divisibly by 24
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Square of any even number is divisible by 4. Square of any odd number divided by 4 gives remainder of 1. So any perfect square can be represented as 4n or 4n+1
- |a+b| < |c+d|, thus (a+b)2 < (c+d)2 -
Using limits to solve inequalities: x2 < 2x < 1/x, x>0 x2 < 2x x < 2 2x < 1/x x < √2/2 x2 < 1/x x < 1 Thus Solution lies 0 < x < √2/2
Sales Price = Cost Price + Profit Sales Price = Market Price – Discount A% of B = B% of A
General Rules
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Number System Dividend = Divisor * Quotient + Remainder Zero Zero is not positive, nor negative. Zero is an even number. Zero is a multiple of every number. Even and Odd Suppose k is an integer. k is an odd number k=2r+1, k is an even number k=2r Useful Facts odd ± odd --> even even ± even --> even odd ± even --> odd odd × odd --> odd a-b is odd--> a+b odd a-b is even--> a+b even
Mixture Problem
Ex: In what ratio should tea 35$/Kg be mixed with 27$/Kg so that the mixture will cost 30$/Kg? Solution: 35x + 27(1-x) = 30(1) Solving for x we get x= 3/8, the ratio is x/(1-x) = 3/5 Or A/B = (30-27) / (30-35) =3/5
Relative Speed Objects are in same direction, relative speed is their sum. Objects are in opposite direction, relative speed is their difference. If the speed of boat in still water is X and speed of the Stream is Y, then the relative speed of the boat with the stream is X+Y and against the steam is X-Y.
Percentage changes where the product is constant: Distance = Speed X Time, Revenue = Units Sales X Unit Price To keep the product constant, if Speed is changed by a% then the time will change a/(1+a) If the speed is increased by 25%, then the time will decrease by (0.25/1+0.25) = 0.2=20% If the speed is decreased by 25%, then the time will increase by (0.25/1-0.25) = 0.3333=33.33% (1+a)(1+b)=1, ie (1+0.25)(1-0.2)=1 ie (1-0.25)(1+0.3333)=1 In the formula a/(1+a) a = a%/100 and is positive if increase and negative if decrease
Inequalities:
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One Equation with two unknowns 15 x + 29 y = 440, this equation will have unique solution only if coefficient of x and y are coprime , and also satisfy the following inequality 29 *15 +smaller coefficient 15 >440 435 + 15 = 450 >440 implies this equation will have unique solution.
Minute And Hour Hand of a Clock The hour hand moves The minute hand moves
degrees in 1 minute. degrees in 1 minute.
Hence at 12:24, after 24 minutes from 12:00, when both hands are vertical, hour hand will move degrees from the vertical position and minute hand will move degrees from vertical position. So the angle between them will be degrees. There is general formula for this, (though no need to memorize):
Percentages: A is 200% greater than B A=3B A=3B A is 200% greater than B
Ratios: a/b = 70/63, how much % is b less than a (70-63)/70 = 7/70 10%
VA/VB = 3/4 thus VA is (4-3)/4 less than B x/y = 8 thus y/x =1/8 and y is (8-1)/8 less than x
3^(3^3) is not 3^(9). It is 3^(27)
Geometry
Circles
The diagonals of a parallelogram bisect each other KN = NM and JN = NL The area of JKLM is equal to 4 × 6 = 24.
cyclic quadrilateral is a quadrilateral whose vertices all lie on a single circle.
Diagonals - In squares and rectangles, diagonals bisect - I rhombus, diagonals bisect at 90 degrees.
Triangle
Formula
Area of Isosceles Triangle Area of Right Triangle
√ (L1. L2)
√ .
Area of Equilateral triangle
Square Area#1
Parallelogram S2
Area #2
Perimeter: 2(L+W) .
Area: B.W
Trapezoid
Rhombus
Perimeter: B1+B2+S1+S2
Perimeter: 4S
Area : 1/2(B1+B2).H
Area #1: B.H Area#2 : (
Circle Circumference(C) Area Diameter Radius
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Formula . . C/ C/2
Area of Sector
. Arc length(Central Angle)
.C Arc length(Inscribed Angle)
.C Rectangular Solids Volume : L.W.H Total Surface: 2(LH+LW+WH)
Cylinder Lateral Surface Area Total Surface Area Volume
Cube Area of Cube: 6 Volume:
Formula . + .
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Coordinate Geometry Distance
Formula
Mid-Point General form
Y = mx+b
Slope Equation of a Line
y-y1 = a(x-x1)
Polygon
Formula
Sum of interior angles of a polygon with n sides = (n-2)*180 Number of Diagonals Nd = n(n-3)/2
Polygons are convex ie there interior angles are less than 180 degrees. Pentagon and hexagon are 5 and 6 sided polygons respectively. Coordinate Geometry Two lines are parallel if their slopes are equal. Two lines are perpendicular if the product of their slopes = -1
1. In case of triangles, an equilateral triangle has maximum area. For a given area equilateral triangle has the smallest perimeter. 2. Triangles of equal heights have areas proportional to their corresponding bases. 3. Triangles of equal bases have areas proportional to their corresponding heights. 4. Areas of two triangles are equal if they have the same base and lie between the same parallel line. 5. In two similar triangles, the ratio of their areas is the square of the ratio of their sides. 6. Angle between two lines:
Parabola Here equation for parabola is form x2 = 4ay
if parabola is offset by(h,k) then equation will be ( X-h)2 = 4a(Y-k)
for the following type of parabola the equation will be Y2 = 4aX
if parabola is offset by(h,k) then equation will be ( Y-k)2 = 4a(X-h)
Circles -
The perpendicular from the center of a circle to a chord of the circle bisects the chord. OM∟AB AM= BM
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Equal chords (CD = AB) are equidistant from the center. Conversely, if two chords are equidistant from the center of a circle, they are equal.
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Two chords of a circle, AB and CD intersect at point P, Then, PA X PB = PC X PD PA2 = PC X PD
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In a circle, equal chords subtend equal angles at the center.
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The angle subtended by an arc of a circle at the center is double the angle subtended by it at any point on the circumference.
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Angles inscribed in the same arc are equal. ̂ ̂
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An angle inscribed in a semi-circle is a right angle. ̂
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The straight line drawn at right angles to a diameter of a circle from its extremity is tangent to the circle.
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If two tangents are drawn to a circle from an exterior point, the lengths of two tangents segments are equal. Also the line joining the exterior point to the center of the circle bisects the angle between the tangents. PA = PB ̂ ̂
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The angle that a tangent to a circle makes with a chord drawn from the point of contact is equal to the angle subtended by that chord in the alternate segment of the circle. ̂ ̂
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When two circles intersect each other, the line joining the centers bisects the common chord and is perpendicular to the common chord.
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In a circle, parallel chords intercept equal arcs. ̂ ̂ arc AC = arc BD
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PQRS is inscribed in a circle. ABCDEF is circumscribed about a circle.
Triangles
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The sum of the two sides of a triangle is greater than the third side.
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The sum of the angles of a triangle is equal to 180°.
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The exterior angle α is equal to the sum of the two opposite interior angles ́ ́
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Area of a Triangle o o
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Medians of Triangle The medians of a triangle are lines joining a vertex to the midpoint of the opposite side. In the figures below, AF, BD and CE are medians The point where the three medians intersect (O) is the centroid.
o The medians divide the triangle into two equal parts o o The centroid divides a median internally in the ratio 2:1 o Apollonius Theorem
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Altitudes of a Triangle
The altitudes are perpendiculars dropped from a vertex to the opposite side. In the figure below, AN, BF and CE are altitudes and their point of intersection is the orthocenter. o ̂ ̂ o ̂ ̂
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Internal Angle Bisectors of a Triangle
In the figure below, AD, BE and CF are the internal angle bisectors of triangle ABC. The point of intersection of these angle bisectors I is the in-center of the triangle ABC ie the center of the circle touching the triangle. ̂ ̂ ̂ ̂ ̂ o ̂ o
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Perpendicular Side Bisectors of a Triangle
In the figure below, the perpendicular bisectors of side AB, BC and AC meet at O the circumcenter (center of the circle passing through the vertices) of triangle ABC. o ̂
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Line Joining The Midpoints of a Triangle
In the figure below, D,E and F are the midpoints of the sides of triangle ABC. o o o
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For Acute Triangles:
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For Obtuse Triangles:
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Right Triangle o Height = BD x AC = AB x BC o Midpoint of Hypotenuse = EA = EB = EC
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General Notes o If BC > AB then angle A > angle C o Sum of any two sides always greater than the third side a+b > c o Difference of any of two sides always less than the third side. a-b < c o The circumradius of a cyclic polygon is a radius of the circle inside which the polygon can be inscribed. o The inradius is the radius of the circle of which the polygon has circumscribed.
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Equilateral Triangles
o The point of intersection of all the 3 bisectors is center of the triangle and is called centroid. o PQ = 1 / 2 BC o Triangles APQ = BQR = QCR = PQR o Equilateral Triangle with side a Height = h= (√3/2)a Area = A = (√3/4)a2 Inradius = a/(2√3) Circumradius = (1/√3)a -
Special Triangles o 45-45-90 triangle has side lengths in proportion to 1-1-√ 2. o A 30-60-90 triangle has side lengths in proportion to 1-√ 3-2. o Right Triangles have sides proportion to 3:4:5 5:12:13 7:24:25 8:15:17 9:12:15
