GRE BOOK 3- Geometry
parallel lines cut by a transversal -transversal is just a line that goes between the two lines and appears to connect them
acute angles= less than 90 degrees obtuse angles= more than 90 less than 180 WHEN SEE THIS ON EXAM -fill in all acute angles -fill in all obtuse angles (sometimes test hides this by making a Z shape) extend out lines to be safe
RECTANGELS specific type of parallelogram
all FOUR INTERIOR ANGLES OF A RECTANGLE are RIGHT angles PERIMETER= sum lengths of the four sides OR add the lengths + width and multiply by 2 AREA= length X width (since are perpendicular dont need height) (if given a rectangle with a hypotenuse to form triangles use the pythogreom therom to find the other sides) **dont assume they are equal unless you see a right angle drawn**
RECTANGLES
all the INTERIOR angles are RIGHT Angles AREA= LENGTH x WIDTH
ANGLES of TRIANGLES
alligator mouth -the tip of mouth is the largest angle -the start of the mouth (inside) is the smallest angle -the opening of the MOUTH is always the SMALLEST angle SIDES match the angles -the LONGEST side= opposite the longest angle -the SMALLEST side= opposite the smallest angle
RIGHT triangle
any triangle one of the angles is a right angle use the PYTHAGOREAN THEROM with right triangles a^2 + B^2 = C^2 A and B are the SIDES C= the hypotenuse the side OPPOSITE the right angle remember when doing the Pythagorean theorem you are always finding SIDES so they will be smaller numbers if you find the square you need to find the square root to find the single digit.
AREA of a triangle
area= 1/2 base X height -the base and the height MUST be perpendicular to each other -height will form a 90 degree angle with the base ANY side can be the BASE of the triangle -the height can also be drawn outside the triangle
DISTANCE BETWEEN TWO points
calculated using the pythagoream therom 1) plot your points 2) draw a triangle connecting the points 3) calculate the RISE and RUN of the two legs of the triangle 4) use the pythoream therom after drawing them out JUST COUNT keep it positive numbers and plug into the pythagreom therom
CIRCLES
circumference= 2pie(radius) circumference= pie (diameter) AREA= pie(radius^2) diameter= 2(radius) the diameter can often been seen STRAIGHT Across but it can also be drawn DIAGNOL which can help you find the hypotenuse of a triangle or a diagonal of the rectangle ALWAYS LOOK FOR RELATIONSHIPS BETWEEN SHAPES.
PYTHAGOREAN TRIPLES -memorize these -first two numbers are the sides -last is the hypotenuse ONLY WORKS ON RIGHT TRIANGLES
common 3-4-5 when squared= 25 key multiples = 6-18-10 or 9-12-15 or 12-16-20 5-12-13 when squared= 169 key multiples= 10-24-26 8-15-17 when squared= 289 key multiples = NONE the key multiples are numbers you can DIVIDE into the smaller main ones
to find the LENGTH of a HYPOTENUSE
length^2 X width^2= hypotenuse^2
QUADRILATERAL -figure with four sides commonly seen are PARRALELLOGRAMS (opposite sides are equal and opposite angels are equal) parallelogram-the diagonal will separate into two equal triangles
parallelograms- must be perpendicular: PERMITER= the sum of all sides AREA= base X height
PERIMETER of POLYGONS
perimeter= the DISTANCE around a polygon the SUM of the length of ALL SIDES (just ADD all sides together)
POLYGONS many questions about triangles will also involve quadrilaterals -must understand: 2D shapes= PERMITER/. AREA/ INTERIOR ANGLES 3D shapes= SURFACE AREA/ VOLUME
polygon= closed shape formed by line segments -triangles- three side shapes -quadrilaterials (four side shapes) N side shapes- other polygons 5+ sides QUADRILATERALS- main focus on GRE -squares / rectangles/ parallelograms/ trapezoids
HOW TO PLOT A LINE from a problem y=1/2x-2
start with Y ALWAYS -start with plotting -2 on your number line figure out how to use the slope M =1/2 in this case -think of EVERY M as a FRACTION 1= numerator= RISE= change in Y ( up and down) 2= denominator = RUN= = change in X (left or right) after you find your spot you can plug in X value and solve for Y to see if it is correct PAY ATTENTION IF YOUR M VALUE IS (+) OR (-)
when dealing with geometry on the GRE
step one= redraw the shapes with the information they want you to find step two= look for math equations from the diagram you have drawn and look for the missing values stated -area -length -width -angles
LINES
straight line =180 degrees parallel lines - are two lines that run across from eachother BUT never meet perpendicular lines= lines that intersect at a 90 degree angle KNOW FOR EXAM 1) angles formed by any intersecting line 2) angles formed by parallel lines cut by a transversal
POLYGONS INTERIOR ANGLES
sum of the interior angles = NUMBER OF SIDES in the polygon triangle= 3 sides= 180 interior angles quadrilateral= 4 sides= 360 interior angles pentagon= 5 sides= 540 interior angles hexagon= 6 sides= 720 interior angles FORMULA for polygons (n-2) X 180 = will give the SUM of the interior angles **if forget break into shapes with KNOWN angle amounts then ADD 180 for each additional side **
TRIANGLES
the diagonal is called a HYPOTENUSE of a RIGHT triangle ALL the four INTERIOR corners of triangles - are RIGHT angles AREA= 1/2 (base) (height)
NUMBER LINE
the open circle means the number is NOT included to right= positive number to left= negative number
SQUARE
PERMITER and AREA= knowing one side is enough to figure out all sides Area= length X width perimeter= adding all sides
finding MAX and MIN AREAS of POLYGONS -general rule- all quadrilaterals with a given PERIMETER the SQUARE has the LARGEST AREA -general rule- all quadrilaterals with a given AREA the square has the MINIMUM perimeter a regular polygon with ALL sides equal will maximize the area for a given permiter and minimize perimeter for a given area general rule -when looking at triangles with sides of 3 and 4 they come in many shapes and sizes LOOK TO PLACE THE two triangles PERPINDICULAR to each other to find the MAX area (area of triangle = 1/2 b(h) and parallelogram = b(h) -to find the biggest height place two triangles perpendicular to eachother to keep the height at 3 units
(might ask to find the max area of a quadrilateral with a fixed peremiter= a very specific number)
CENTRAL ANGLE angle whose vertex lies at the center point of a circle -defines both ARC and SECTOR of a circle
-the degree measure between two radii sector area/ circle area= arc length/ circumference HOW TO CALCULATE -remember to keep SECTOR information from ENTIRE CIRCLE information every question will help you start by finding one of these three things -use this information to find other information 1) central angle/360 2) sector area/ circle area 3) arc length/ circumference 1=2=3 **compare each two to form an equation by setting them equal to each other**
QUICK TIPS
-when given a word problem and their is a PATH a person takes -DRAW out path to find the geometry shape (add up same direction together for a total length) -when solving triangles LOOK for the RIGHT angle symbol to see if it can be a 90 degree combination of common shapes -any time see a RIGHT triangle and one sides has a length of a multiple of square root of 3 CHECK if it is a 30-60-90 triangle
AREA of a figure is
AREA= LENGTH X WIDTH
AREA of POLYGONS
AREA= the space INSIDE the polygon -measured in square units (cm2) or (m2) (ft2) MUST MEMORIZE -AREA OF A TRIANGLE= BASE x HEIGHT/2 -height always refers to a line that is perpendicular (long tall angle) at a 90 degree angle to the base AREA OF A RECTANGLE = LENGTH x WIDTH AREA OF A TRAPEZOID= BASE 1 + BASE 2/2 x HEIGHT -height refers to a line perpendicular to the two bases which are parallel (may have to draw the line in) (parallel means opposite sides) Or you can average the TWO bases and multiply by the height to GET THE SAME answers AREA OF A PARALLELOGRAM= BASE x HEIGHT
OTHER POLYGONS -right triangles can help find other polygons squares cubes rectangles rectangular solids
FORMULA to find the DIAGNOL of a SQUARE d= S square root of 2 (s = the side of the square or the diagonal of a face of a cube) ANY SQUARE- can be divided into -45-45-90 triangle -ratio 1:1: square root of 2 Pythagorean theorem DIAGNOL OF A CUBE -to find the main diagonal of a cube D= S square root of 3 (s is the EDGE of the cube) RECTANGLE -to find the diagonal of a rectangle know either the LENGTH or WIDTH or one dimension and the proportion of one to the other -use the Pythagorean theorem a2 + b2= C2
TRIANGLES (all over GRE)
RULES: -the SUM of ANY TWO side LENGTHS of the triangle will ALWAYS be GREATER than the third side length (take the number given add to your side to see if it is LESS than the other number side given in the equation) -to find the biggest and smallest values of the triangle take TWO sides and ADD them get your answer then also SUBTRACT them to get your answer answer < X < answer the INTERNAL angles of a triangle MUST ADD up to 180 degrees - this can be given as X and 2X so find X first then find 2X to get the other angle DONT try to find angles by picture ALWAYS SOLVE MATHMATICALLY if given a STRAIGHT line with a DEGREE mark - remember ALL STRAIGHT LINES have a measure of 180 degrees so find the line with the X angle that is missing by basing it off of 180
CYLINDER -made of two circles and one rectangle
SURFACE AREA of a cylinder -sums of each shape -circle A= pie(r^2) -circle A= pie (r^2) -rectangle A= LX W OR SA= 2 circles + 1 rectangle = 2pie(r^2) + 2pie(r)(h) overall KNOW 1) the radius of the cylinder 2) height of the cylinder
POLYGON -QUADRILATERALS
TRAPEZOID -one pair of opposite sides are parallel (2 sides are equal) (2 sides are NOT equal) -angles are NOT EQUAL -lengths are NOT EQUAL PARALLELOGRAM and RHOMBUS -some angles are equal -some lengths are equal parallelogram= opposite sides and angles are equal -rhombus- opposite sides and opposite angles are equal = ALL SIDES ARE EQUAL RECTANGLE and SQUARES -rectangle- all angles are 90 degrees / opposite sides are equal -squares= ALL angles are 90 degrees / ALL sides are equal
INSCRIBED ANGLE -vertex is on the circle itself (has a longer angle touches the line of the circle (not in center)) some can have both a central angle and a inscribed angle -check DIAGRAMS carefully
a inscribed angle = half of the arc it intercepts can see INSCRIBED TRIANGLES in circles- all vertices will have points on the circle -if one of the inscribed triangle sides is a DIAMETER of the circle then the triangle must be a RIGHT TRIANGLE
exterior angles of a triangle
equal in measure to the SUM of the two non-adjacent (opposite) interior angles of the triangle
CIRCLES
in a circle there a set of points that are all the same distance from a central point RADIUS- the distance between the center of a circle and a point on a circle RADII- are line segments from the center (O) to a point on edge of a circle (r) all radii in the same circle have the same length DIAMETER- a line drawn across the center of the circle connecting to edge of circle points -always 2X the length of the radius (d= 2r) CIRCUMFRENCE- measure of the distance around a circle pie X d= circumference pie=3.14 (always) AREA= the space INSIDE the circle A= pie r^2
INTERSECTING LINES
interior angles formed by interesting lines form a circle = 360 degrees supplementary angles= interior angles that combine to form a line 180 degrees vertical angles -angels found opposite each other where the two lines intersect are EQUAL HOW TO DO THESE PROBLEMS -if given two values on the same side that equal less than 180 and missing value find it then compare it to the answer it is asking for- by checking if it is supplementary/ vertical
SECTOR of a circle
is a fractional PIECE of the circle (slice of the circle) with sector cut in HALF - the DIAMETER and RADIUS STAY THE SAME -the AREA and CIRCUMFRENCE are now HALF of what they were ARC LENGTH= the portion of the circumference that remains (circumference of a semicircle also has to get cut in half )
the PERIMETER
is the sum of ALL LENGTHS of ALL three sides -when trying to solve perimeter keep in mind the type of triangle you have check angles and sides to see if they are equal.
intercept
is where the line intersects a coordinate axis X intercept= on X axis y intercept = on Y axis
SURFACE AREAS 3 dimensions -rectangular solids -cubes BOTH of these shapes have 6 faces so you are finding the AREA OF ALL SIX FACES!
the surface area of 3D shape the amount of space on the SURFACE of that object commonly measured in square inches or square feet Surface area= the SUM of the areas of ALL of the faces RECTANGLUAR SOLID= **take all known sides and multiply by 2 for a total of 6 sides** CUBES- all sides are equal lengths find AREA of one face # x # then multiply your answer by 6 for the TOTAL area
EQUILATERAL triangle
triangle with THREE equal angles and three equal sides each angle of a an equilateral is 60 degrees= ALL THREE angles must add up to 180 degrees 30-60-90 triangles= if have TWO of these they can make up a equilateral triangle FORMULA RULES for 30-60-90 triangle -short leg= 1 or X long leg= square root of 3 or x square root of 3 hypotenuse = 2 or 2X (always try to find the length of SHORT leg if can)
ISOCOLES triangle
triangle with TWO equal angles and TWO equal sides ISOCOLES RIGHT TRIANGLE -a 90 degree angle opposite the hypotenuse and two 45 degree angles = 45-45-90 triangle the right triangle FORMULA has a multiple of these sides 45= leg 1 or X 45= leg 1 or X 90= hypotenuse square root of 2 or X square root of 2 TWO of the right triangles make up a SQUARE if you are given a diagonal of a square you can find the 45-45-90 ratio to find the length of a side of square
LINE RELATIONSHIPS
use the equation Y=mx + b m and b= numbers (positive, negative, zero) when given a problem y=5x +3 M= 5 B= 3 a negative in place of M= a negative 1 ALL of the above are linear equations NOT LINEAR EQUATIONS y= x2 y= 1/x M= SLOPE tells you how steep the line is if line is rising or falling rising= positive quadrant to right falling= negative quadratn to left and down steep= close to the axis (0) gentle slope is right above the x axis B= Y INTERCEPT -tells you where the line crosses the y-axis when X is at ZERO -to find y-intercept plug in 0 for the X equation
volume of a cylinder
volume of a cylinder is how much STUFF it can hold V= pie(r^2) (h) know the 1) the radius of the cylinder 2) the height of the cylinder think of this as a STACK OF CIRCLES -just multiply the pie(r^2) X height of the shape = AREA
VOLUME of 3D shapes -rectangular solid -cube VOLUME= LENGTH x WIDTH x HEIGHT
volume= the amount of STUFF it can hold -seen as cubic inches or cubic feet rectangular solid= all dimensions are different values cube- all dimensions are the SAME
COORDINATE PLANE
x-coordinate is the horizontal line # right= positive # left= negative y -coordinate is the vertical line #on top= positive # below= negative (#=X, #=y) is short hand version a line= one dimensional a plane= two dimensional QUADRANTS move from right top -->left top-->bottom left--> ends in bottom right 1= positive x and y 2= negative X and positive y 3= negative x and y 4= positive X and negative Y READING CURVES on a plane -start with the area you know and move up till reach the curved line and that is the other coordinate