Geometry Flashcards
Isoleces right triangle hypotenuse
SQRT(2) * side
30-60-90 triangle base form
x | SQRT(3)x | 2XC
Area of equilaterals
[ Side² * SQRT(3) ] / 4
What are the 5 types of quadrilaterals
Paralellogram, Rhombus, Rectangle, Square, Trapezoid
How to calculate the hight of the equilateral triangle?
h = (side * sq3)/2
Descibre the properties presented in a equilateral triangle and a inscribed and circunscribed circle
Inscribed cicle: radius is h/3
What does the diagonals of trapezoid create
It creates two SIMILAR triangles
What is the approach to geometry questions?
Draw the figure with every info you have. Look at the whole figure (from specific to general level) Infere everything you can.
Then, take each assumption and evaluate how you can use it
What is the result of an odd divided by X?
The result is odd if it’s an integer
How to quickly solve fractions divided by 25 and 125?
Multiply by 4 to get to 100 or by 8 to get to 1000
What is the approach for test cases?
1) note what is obvious about the constraints (I.e. Must be positive)
2) think of how you could create an alternative case
How to represent a set of consecutive integers?
I.e. A set of 5 elements: S-2, s-1, s, s+1,s+2
Average = Med = S
How to calculate direct and inverse proportions?
Direct: multiply by X
Inverse: multiply by 1/x
Think of D=V*T (directly proportional) and D/V=T (inversely proportional, time increases as speed reduces)
What is one approach to solve a system of inequalities?
Combine inequalities if they have the same direction
What is one approach to visualize and solve inequalities?
Plot a number line to compare numbers, I.e. 4x>3y
What can you Infere from x/y<1?
And from x/y>1?
X/y<1: nothing, as it could be positive or negative, could be greater on top or bottom
X/y>1 either positives with x>y, or negatives with x>y
What did you learn with the exercise about remainder of 20/P?
That if it takes too many calculations to prove/disprove a DS, it’s either 1) not sufficient, 2) a math property, or 3) you should look for a specific case that disproves it
What is the equation of the ovelarpponf sets?
Of three: A+B+C - (AB + BC +CA - d) + E = T [for greater intersection]
Or A+B+C - (x + y + z + 2d) + E = T [for limited intersection]
What did you learn with the DS question asking the value of 9^p * 27^3r?
1) translate the question: simplify the expression
2) make the statements as similar to the question as possible
What is the exception case for SQ(X)?
If 0<x<1, SQ(x) is greater than x
i.e. SQ(0,5) = ~0,7
If SQ(x)>z, doesn’t mean x > z
What is the last check on Absolute equations/inequalities questions?
Plug the value back into the absolute equation and see if it holds true
|1-x| > 0
|1-x| = -5 - 3 doesn’t exist
Given x>y>z, which could be true?
1) x³>y³>z³
2) z>y³>x³
3) x³>z³>y³
First of all, questions like this are always testing the exceptions
2) is true for near decimals, 0,9 > 0,89 > 0,88
3) is not true as signal is not changed and z is always more impacted than y
What is the range of a set of N consecutive integers?
Ragen is n-1
R = (S+n-1)-S = n-1
Note: a set with repeating numbers could also have range=n-1
What did you learn with the question about two consecutive positive integers divided by 25?
26/25 leaves R=1;
Evaluate each calc with care, as it is designed to make you miss. First case, 24 and 25 left remainders = 24; Second case, 25 and 26, left remainders = 1; which is not even
What was the trick:
“Each stroke of a gammer inserts 30% of the nail, out of the wood, into the wood.”
When testing cases, what to look for?
Mindset: you’re testing RANGES/Behaviors
IESD: Integers, exponentes, signals or decimals
Integers even or odd
Exponents even or odd
Signals + and -
Decimals between -1 and 1
Whats the area of a rhombus?
a = (D * d ) / 2
What is the sum of the positive multiples of 6 that are less than or equal to 714?
Elements: 714/6 = 119
Max + Min = 714 + 6 = 720
(720/2)*119 = 42.840
(AVG max&min)*qtt
A total of 30 percent of the geese included in a certain migration study were male. If some of the geese migrated during the study and 20 percent of the migrating geese were male, what was the ratio of the migration rate for the male geese to the migration rate for the female geese?
Ratio: (20%x / 30) / (80%x / 70)
If X²=4x, what is the value of X?
X = { 0 , 4 }
How to solve questions with too much calculation?
I.E. (3630 * 3800 ) / 43560 * 30
Find common factors (i.e. 12)
Simplify the formula, step by step (i.e. dividing by 12)
Be dilligent about each step to not confuse your calculations
I.E., 43560 / 12 = 3630
What is the square of the difference?
(a-b)² = a² + b² - 2ab
What are trickier ways to hide difference of squares?
Higher exponents (ex 4)
Base equal to 1
Quadratics without exponents (ex 16)
How to compare fractions with cross multiplication?
A/B vs. M/N
Which is larger, AN vs BM?
A*N represents A/B
What are the conditions for terminating fractions?
Numbers composed only of 2 and/or 5
If it has any other factor than you need to test
What are multiples of 7 and 21 that look like prime?
91 = 13 * 7
119 = 17 * 7
133 = 19 * 7
143 = 13 * 11
How to get from the decimal part of a number to the remainder?
Decimal part = remainder / divisor
What is the formula of the PA?
An = A1 + K (n-1)
What is the formula of the PG?
An = A1 * K^(n-1)
How does the difference of squares of consecutive integers grow?
It grows by 2
3²-2² = 5
4²-3² = 7
What can you tell about mean and median in symmetrical lists?
Mean and median are the same in symmetrical lists (ex consecutive numbers, etc)
What happens to the standard deviation if you add the same number to each element?
The std dev stays the same if you add the same number to each element, because you maintained the relationship between average and deviation
What is the math for the OR probability case?
P(A) + P(B) - P(A and B)
If events don’t overlap, then AND is zero
What do you do when you see “at least one” on a probability question?
Calculate the probability of that event not happening
What to do to keep structure on even odd questions?
Break down the requirements (ex if pq must be even or odd, then each must be…)
Think of the conditions (p or q must be even)
What is lcm and gcf?
LCM: the smallest in common number that A and B can be multiplied to get to
GCF: from all factors of each, the greatest number in common
What did you learn with the clock question?
For questions with schemes (ex clock hours, time passing, opposing directions) draw the figures to understand the relationship and the ask
What is an ABCD polígono?
Clockwise, the edges of the polygon are A, B, C, d
If x * sqr(5) is an int, is x an int?
Not necessarily
X = root 5
X = 0
What did you learn with the 30!/10! question?
Read as an executive - attention to detail
“product of CONSECUTIVE INTEGERS”
1 * 11 * 12 (…) is not consecutive
How many multiples of 2 or 25 between 101 and 1000?
Multiples of 2: 1000-102 / 2 ) +1
Odd mult of 25: ( (975 - 125) / 50 )+ 1 = 18
Careful: starts on 101: total is 1000 - 101 + 1 = 900
What is the relationship of time and work between two machines?
Ratio of Time is the reciprocal of Ratio of Work
IF A does Z in X, and B in Y,
then A produces Y in the same time B produces X
Time: X/Y, Work Y/X
Share of A: Y / (X + Y)
What did you learn with the question on 15 houses sold, med 130 and avg 150?
“At least one of the homes was sold for more than 165”
Maximize the lower half to minimize the upper half and see if the average is less than 165
Max(lower) = 208 = 160 gap
Min(upper) = 157 = 105 delta
105<160 -> doesnt meet requirements
at least one house is greater than 165
What is the catch with (x-y) (x+y) = 7 ?
Product of two numbers equals Prime;
One factor is 7 and other 1
What is given from Z ^ (n+1) = Z ^ (n +5) ; for all N?
Z is -1 or 1, because 1 is the only based to any power that equals itself
What did you learn with the question about students preference of M and N?
You can group No/not sure in one group to calculate how many did not like either
Then, you set up a set analysis of which answered Yes, and see the overlap to calculate the Outside
What did you learn on the sets question of garden and lobby?
If the middle is equal to the outside, and you know garden, you can calculate lobby (mentalize bitch)
What did you learn witht he question of seven numbers with MED 63 and AVG 49?
Minimize LHS (x, x, x,), minimize middle (63, 63, 63) , and sum everything (7 + 4x) to get to total (7 * 49)
