Geometry Flashcards
Angles that form a straight line measure…
180 degrees
A transversal forms how many angles ?
8
Describe a transversal
A line that intersects two or more parallel lines
What are the two distinct angles a transversal forms ?
A big angle larger than 90 degrees. And a small angle smaller than 90 degrees
What is the sum rule to remember for transversals ?
The sum of any big angle and small angle formed by a transversal is 180 degrees
What should you always do when working with a problem with a transversal?
Label all 8 angles so that angle relationships are easier to distinguish
What do you call a line that splits an angle or line segment in half
A bisector
What is a midpoint?
Any point that lies in the middle of a line segment
How do you approach a number line problem?
Label the line segments
Set up equations and Rewrite the given information algebraically
Plug the equation into one another
What is a polygon?
A closed figure formes by 3 or more line segments
What do you call the line segments of a polygon?
Sides
What is a point of intersection between two sides called
Vertex
What do you call the distance around a polygon
Perimeter
How do you calculate perimeter of a polygon?
Add the lengths of its sides
The area of a polygon is measured in terms of
Square units
What do you call angles located within a polygon?
Interior angles
Number of interiors angles is equivalent to …?
Number of sides
What is the formula for the sum of the interior angles of any polygon?
(N-2)180= sum of the interior angles of a polygon
Where n is equal to the number of sides a polygon contains
What is the sum of an interior angle and an exterior angle?
180
Because together they form a straight line
What is the sum of the exterior angles of a polygon ?
360
What is a diagonal ?
Any line within a polygon that extends between two vertices but is not a sideshow
What is the formula for the number of diagonals
[n(n-3)] / 2
Where n= the number of sides in the polygon
Why should you always label the interior angles of a polygon?
The angles of a polygon always add to a fixed sum
When visualizing shapes it is important to…?
Recognize every possible shape
When working with angles be sure to watch out for …
The shape that has all of its angles defined as it is usually key to solving the problem
Exterior angles are equal to the sum of …
Their two opposite interior angles
When there are multiple shapes always choose which one?
The one in which all angles are labeled
What is the side opposite a right angle in a right Triangle
Hypotenuse
What are the three sides of a right triangle called?
Hypotenuse, leg , leg
What is the measurement of each of the three angles in an equaltiral triangle
60 degrees
What do you know about an isosceles triangle ?
It has two sides of equal length
The angles opposite the equal legs of an isosceles triangles are also equal
What is the pythagorean theorem?
a^2 + b^2 = c^2
Only used for right triangles
How do you solve a sexy pythagorean ( right triangle w long legs ) problem?
x^2 - y^2 = (x+y)(x-y)
Make the unknown side
x^2= c^2 - b^2 = (c + b) (c-b)
What are some examples of common pythagorean triples?
3-4-5 [6-8-10, 9-12-15, 12-16-20 , 15, 20, 25]
5-12-13 [10-24-26]
8-15-17
7-24-25
What do you call a right triangle that has two angles or sides of equal measure
Right isosceles
What are the angle measurements of a right isosceles triangle
90 , 45 , 45
What are the ratios of the sides of a right isosceles triangle?
x: x : x(sq rt 2)
What is the short cut to determining the legs of a right isosceles triangle if the hypotenuse is not written as x sqr rt 2
Divide it by 2
Multiply it by sqr rt 2
I.e. if c = 10
Legs = 5 sqr rt 2
What is the ratio of the lengths of the legs of every 30-60-90 triangle?
x : x sqr root 3 : 2x
If the length of the hypotenuse of a right triangle doubles the length of one of its legs then it is a ….
30-60-90 triangle
Two 30-60-90 triangles make …
An equilateral triangle
Twelve 30-60-90 triangles make up
A hexagon
Every right triangle in a problem is usually
A special right triangle
What is the correlation between the sides and angles of a triangle?
The smallest angle of any triangle is always across from the smallest side
The largest angle of any triangle is always across from the largest side
If two angles are equal the…
The opposite sides are equal
What is the triangle inequality theorem
For a triangle to exist, The length of a given side must be less than the sum of the other two sides but greater than the difference between those two sides
What are the characteristics of a similar triangle?
Equal angles
Or
Proportional sides
What are the corollaries of similar triangles ?
If 2 angles of any 2 triangles are equal, the third angle must be as well
If the angles of any 2 triangles are equal, the sides of those triangles are proportional
If the sides of any 2 triangles are proportional, the angles of those triangles are equal
What is the advanced trick for similar triangles when it comes to area ?
The ratio of the areas of two similar triangles is equal to the square of the ratio of the corresponding sides
I.e. if a triangle is two times as big as another its area will be 2^2 or 4 times as large as its similar triangle
What is the formula for area of a triangle ?
(1/2) X Base X Height
Whats another’s name for height
Altitude
How can you calculate the area of an oddly shaped triangle w/o a clear base or height?
Consider rotating the triangle so that it has a defined horizontal base
… of a triangle is defined as the side that runs perpendicular to the height
Base
…of a triangle is defined as the perpendicular distance from a vertex to the side opposite that vertex
Height (altitude)
Advanced note…
The area of a triangle with sides x and y gets larger as the angle between legs x and y gets closer to 90 degrees
What is an exterior angle?
Any angle formed by extending a side of a triangle .
The addition of an interior angle and an exterior angle is…
180 degrees
An exterior angle is equal to the sum of its …
Two opposite interests angles
What does a triangle with parallel lines signify ?
Similar triangles
If a larger right triangle is split into two smaller tight triangles then…
All three triangles are similar
If the hypotenuse doubles the leg you are working with a … special right triangle
30-60-90
Diagonals of a rhombus always cross at what degree?
90
If two sides of a triangles are the same then ?
There corresponding angles are the same
If you are trying to maximize the area of a triangle you should?
Make it a right triangle
If possible make it a right isosceles triangles
What happens when you cut an equilateral triangle in half?
You get two 30-60-90 triangles
In terms of triangles, what is a hexagon made of?
6 equilateral triangles
What is the ratio of perimeter and area for similar triangles ?
The ratio of the area is the square of the ratio of the perimeters
One big right triangle cut into two right triangles makes for …
Two right triangles that are proportional (similar) to the larger 1
Anytime there is a parallel line drawn in a triangle there is a ?
Pair of similar triangles
What is the sum of the interior angles of a quadrilateral?
360
What are the five special quadrilaterals
Rectangles Squares Rhombuses Trapezoids parallelograms
This is a quadrilateral with opposite sides of equal lengths and 4 90 degree angles
Rectangle
Describe the diagonals of a rectangle
They are equal
They cut each other into 4 equal halves
Do NOT intersect one and other at 90 degree angles
Do NOT bisect the corner angles of a rectangle
A square is made up of what triangles?
Four 45-45-90 triangles
Describe the diagonals of a square
They are equal
They intersect at a 90 degree angle
They bisect the corner angles
What is the formula for the area of a rectangle and square?
Length x width
A rhombus can be described as an?
Equilateral parallelogram
Slanted squares
What do you call a slanted quadrilateral whose opposite sides are parallel and equal in length?
Parallelogram
Do the diagonals of parallelograms and rhombuses bisect each other ?
Yes
Are the bisections of parallelograms and rhombuses equal in length?
No
The diagonals of a rhombus split it into?
4 right triangles as they meet at 90 degree angles
What are the angle rules for parallelograms and rhombuses?
Opposite angles are equal
Adjacent angles add up to 180
What is the formula for area of a parallelogram and a rhombus ?
Base x height
What is the formula for area of a rhombus
(Diagonal 1 x Diagnol 2 )/2
A quadrilateral with one set of opposing sides that are parallel but unequal in length?
Trapezoid
What is the formula for area of a trapezoid ?
[(Base 1 + Base 2) / 2 ] x height
To maximize the area of any quadrilateral what shape should you make it?
A square
How do you minimize the perimeter of a quadrilateral? What shape should you use?
A square
What should you remember about angles of rhombuses and parallelograms ?
Opposite angles are equal and adjacent angles add up to 180
Is a square a parallelogram?
No because its sides aren’t slanted
A diameter of a circle must do what ?
Pass through the center of a circle
Be twice the length of a radius
Touch two points of a circle
What is a chord?
Straight line drawn from one point on a circle to another
What is a tangent?
Straight line drawn outside a circle that intersects the circle at a single point
What is the longest possible chord of a circle?
Diameter
What is to be known about a chord that runs perpendicular to a radius?
It will be bisected by that radius
What is to be known about a radius that is drawn to a point of tangency?
It intersects the tangent at 90 degrees
What is the formula for area of a circle?
Pi r squared
What is the formula for a circumference of a circle?
2 pi r
Or
Pi d
What should you use when a question ask you to approximate the circumference or area of a circle
Approximate pi as
3 or (22/7)
What is a portion of the circumference of a circle called?
An arc
What is the portion of the area of a circle known as?
Sector
What us a major arc?
It is an arc greater than half the circumference of a circle
What is a minor arc?
An arc that is smaller than half the circumference of a circle
What is needed to find the length of an arc?
Circumference and central angle
What is the formula for length of an arc?
2 pi r * (x/360)
All sectors must…?
Extend from the center of the circle
What is the formula for area of a sector?
Pi r squared * (x/360)
What is an inscribed angle?
Any angle whose vertex lies on the perimeter of a circle
What are the three properties of inscribed angles in circles?
The inscribed angle of an arc is always half the measure of its central angle
Inscribed angles drawn to the same arc have equal measures
Inscribed angles of equal measure have arcs of equal length, and arcs of equal length have inscribed angles of equal measure
What are the two properties for inscribed and circumscribed polygons?
An inscribed or circumscribed polygon always has the same center as the circle it inscribes or about which it is circumscribed
If a triangle is inscribed in a circle so that one of its sides is a diameter of the circle then that triangle is always a right triangle
What should you always do when working with circles ?
Label your radii
All radii in the same circle have the same…?
Length
What is the surface area of a cylinder formula?
2pirh + 2(pir^2)
Remember its circumference of a circle time height + 2( area of a circle)
How do you calculate the surface area of a band of a cylinder?
2pir*h
What is the highest point of the cone?
Apex
What makes a cone a right circular cone?
The highest point runs perpendicular to it its base
What do you call half of a sphere
A hemisphere
What is there to know about the cross sections of a sphere?
They are circles
The closer the cross section is to the middle of a sphere the larger it is
What is a 3-D figure w triangle sides and a polygon base?
Pyramid
How do you solve any problem in which the specific dimensions of a 3D shape are unquantified
Pick numbers that would fit the formula and give you the answer for whats given
What volume question should you be aware of ?
A volume question asking how many 3d figures can fit inside another
The answer is : UNkNOWN
You must know the exact dimensions of each shape
How do you calculate the volume of a cylinder ?
Area of a circle times height
How do you calculate slope?
Slope= (y2-y1)/(x2-x1)
I.e. the difference between y-coordinates over the difference in x coordinates
What is an intercept?
Any point where a line intersects a coordinate axis
When working with coordinate geometry it is critical that….
Every equation be rewritten in the form of y=mx+b
What is the y intercept of equation such as y=3x
0,0
What is the slope of an equation such as y=x +2
1
How do you find the x intercept of a line ?
-b/m
Or plug 0 in for y and solve for x
Whats a quick way to find the x or y intercept?
Plug 0 in for x to find the y intercept
Plug 0 in for y to find the x intercept
How do you graph linear inequalities?
Put the equation in y=mx + b form
Pick a point (its easy to pick 0,0) to plug into the inequality
If the statement is true shade the side of the line that contains the point , if its false shade the side that doesnt contain that point
Whats the slope of a horizontal line ?
0
Whats the slope of a vertical line ?
Undefined
What is the equation form for a horizontal line?
Y=b
Whats the slope of a vertical line ?
Undefined
What is the equation of a vertical line ?
X= a
Where a is the x intercept
What is important to remember about parallel lines?
They do not intersect
They have the same slope
What is important to remember about perpendicular lines?
They intersect at a 90 degree angle
The slopes of two perpendicular lines are opposite reciprocals i.e. their product is -1
How do you find the distance between two points ?
Find the slope of the line So you know the rise and run
In vision a right triangle and the line is the hypotenuse
Sketch out the two legs to determine the hypotenuse using Pythagorean theorem or special right triangle knowledge
How do you find the midpoint of two coordinates ?
The average of the x and y coordinates
[(x1+x2)/2] , [(y1+y2)/2]
How do you find the equation of a line given two points ?
Find the slope of the line
Then plug any of the given points into y=mx +b using the slope you found
Solve for b
Then you have your equation
How can you establish whether a line and a point intersect ?
Plug the coordinates values of the point into the equation of the line .
If a true statement it is on the line .
If false the statement is not on the line
How can you find the point at which two lines intersect ?
Set the two equations equal to each other and solve for x
Plug the value you got for x into either equation and solve for y
The x and y are your coordinates
What should you remember About a graph that marks one point labeled , what should you look for?
See of the origin is a point
What does a perpendicular bisector do?
Splits a line segment in half while intersecting the segment at a 90 degree angle
If you have two points then, you can find the…?
Slope
How do you find the equation of a perpendicular bisector ?
Determine the slope of the bisected line segment using its two points
Determine the slope of the perpendicular bisector
Plug that slope into y=mx+b
Find the y intercept of the line
Plug everything into y=mx+b
What type of equation is typical for a parabola ?
A quadratic equation
What does a positive a term in a parabolas equation tell you ?
A negative b term ?
Positive = upward parabola Negative = downward parabola
Whats the different parts of a parabalas equation tell you
Y=ax^2 + bx + c
a=slope
b= position relative to the y axis
c= y-intercept
How do you find the x intercepts for a parabola ?
Set the equation equal to 0 and solve for x
If given an equation for a parabola plug in x values so you can find points on the graph
…
What type of parabola results from an equation in the following form x=ay^2 + by + c
A horizontal parabola
What does the “a” in the y ^2 term tell you about a horizontal parabola ?
If its positive it will open to the right
If its negative it will open to the left
What does the “c” term tell you about a horizontal parabola ?
Its x intercept
How do you find the y intercepts of a horizontal parabola?
Set the equation equal to 0 and solve for y
Whats a convex polygon?
A polygon in which each interior angle has a measure less than 180
