Geometry Flashcards
arc length equations
central angle/360
sector area/circle area
arc length/circumference
circumference of a circle
pi X d
or pi X 2r
area of a circle
pi X r^2
diameter of a circle
d=2r
if central angle is 60 degrees
central anlge/360- the whole of the circle
so do 60/360=1/6
so a sector with the central angle of 60 is 1/6 of the entire circle
secto area=
sector area= Fraction X Entire area
so in the ex above, if the original circle has an area of 36pi and diameter of 12
1/6 X 36 pi = 6 pi
arc length=
arc length= Fraction X entire circumference
so in the ex above, if the original circle has an area of 36pi and diameter of 12, so entire circumference is 12 pi
1/6 X 12pi = 2pi
a sector has a radius of 9 and an area of 27 pi. What is the central angle of the sector?
area of whole circle is pi r^2,r=9 so 81pi
sector area/circle area= 27pi/81 pi
27pi/81pi= 1/3
the sector is 1/3 of the entire circle. The full circle has an angle of 360 so multiply 1/3 by 360 to find the area of the sector= 1/3 X 360= 120
sum of any two sides > third side
the sum of any two side lenghts of a triangle is always greater than teh third side length
A related idea is that any side is greater than the difference of the other two side lengths so for example a triangle has side lengths 3 and 5
so less than 8
the third side= x must be less than 3+5 and greater than 5-3 so greater than 2
so 2<x<8
Two sides of a triangle have lengths of 5 and 19. Can the third side have a length of 13?
No the two known sides of the triangle are 5 and 19. The 3rd side of the triangle must be greater than 19-5, and less than 19+5. So the third side x must be greater than 14 and less than 24, number 13 is less than 14 so 13 cannot be the length of the 3rd side
19-5 < x < 19+5
14 < x < 24
Two sides of a triangle have lengths 8 and 17. What is the range of possible values of the length of the third side?
x must be greater than 17-8=9 and less than 17+8= 25
17-8 < x < 17+8
x must be greater than 9 and less than 25
sum of three angles
the sum of internal angles of a traingle must be 180 degrees!!!!!
SUM of three angles = 180
a straight line always has a measure of…..
180
a line always equals 180 degrees!
complementary angles around a line always add up to 180!!!
x/x would be 180 =2x
on quant questions the figures will not say they are drawn to scale, so assume the are not
on problem solving questions a figure will be drawn to scale unless there is a note that sys it is not drawn to scale
same side= same angle
-the longest side is opposite the greatest angle and the shortest side is opposite the smallest angle
-like an alligator, as the angle between it supper and lower jaws increases, the distance between its top and bottom teeth inc
isosceles
a triangle has two equal angles and two equal sides
equilateral triangle
a triangle has three equal angles all 60 degrees and three equal sides
all 60 degrees!
If you see two equal sides in a triangle
If you see two equal angles in a triangle
Set the angles opposite each side equal. The two sides both equal 8, so the angles opposite those sides are equal
Set the sides opposite each angle equal. Two angles equal 30 degree so the sides opposite those angles are equal
the perimeter of a triangle
What is the perimeter of a triangle with sides 5, 8, and 12?
it is the sum of the lengths of all 3 sides!
5+6+10=21 perimeter
5+8+12 = 25 perimeter
Area of a triangle
area of a triangle= 1/2 X base x Height
the base and the height must be perpendicular to each other
pythagorean theorem
a^2 +b^2=c^2
this can only be used for right triangles!! so can use P theroem to find the length of the third side of a right triangle if you know the lengths of other two if right triangle. just has to be a right triangle
3:4:5
6, 8, 10
5, 12, 13
quadrilaterals
is any figure with four sides
they can always be cut up into two triangles by slicing across the middle to connect opposite corners. Therefore what you know about triangles could apply in a problem involving quadrilaterals
difference between quad and parallelogram: As the name suggests, a quadrilateral is a polygon that has 4 sides.
While on the other hand, a parallelogram is a special quadrilateral in which both pairs of opposite sides are parallel and equal.
A parallelogram is a special type of quadrilateral.
A parallelogram is defined as a quadrilateral in which both pairs of opposite sides are parallel and equal.
So, all parallelograms fall under the category of quadrilaterals, but all quadrilaterals can not be named as parallelograms.
parallelograms
any four sided figure in which the opposite sides are parallel and equal. Opposite angles are also equal and adjacent angles, angles that are next to each other without another angle in between, add up to 180!!!
in a triangle ABCD, AB=CD are parallel and have equal lengths, BC=AD are parallel (opposite side remember) and have equal lengths
Angles ADC and ABC are equal, angles BAD and BCD are equal
The diagnol divides the parallelogram into two equal triangles, they also cut each other in half (besides each other)
-Last make sure for area base x height, need to make sure height is perpendicular to base and draw a line down can’t take a side not perpendicular
parallelograms 2
opposite sides are parallel and equal!
opposite angles are also equal adn adjacent angles
they always have two sets of equal sides. for ex if two of the sides have a length o f6 and two of the sides have a length o f8. Therefore, the perimeter equals 2x6 + 2x8= 12+16=28
you need to “drop a height” for base x height area or draw a perpendicular line for the height
rectangles
rectangles are a specific type of parallelogram, they have all the properties of parallelograms but also all four internal angles of a rectangle are right angles*****
one side is the width the other is the length
rectangles= parallelogram + 4 right angles
the diagonal of a rectangle cuts the rectan gle into two equal right triangles!!!! with all the properties that are normal for right triangles
square
square= rectangle+ four equal sides
square 2
the most special kind of rectangle is a square, it is a rectangle in which all four sides are equal
if you know one side of a square is enough to determine the perimeter and area of a square
x=-3
then you know the point can be anywhere along the vertical line at -3 that crosses the x axis , every point on the dotted line has an ex coordinate of -3
if x=3, then a vertical line runs through the number 3 on the x-axis. The y-coordinate could be anything we do not know, you only no one coordinate
y=1
every point on dotted line has a y coordiante of 1, horizontal line, these points form a horizontal line
y=mx+b
y=3x-2
y=-x+4
y=1/2x
m=3, b=-2
m=-1, b=4
m=1/2, b=0
m=
slope of a line
slope tells you how steep the line is and whether the line is rising or falling
m>0 positive slope
m<0 negative slope
m>1 positive slop, very steep
0<m<1 shallow slope
slope change in y/change in x OR rise/run
b=
y intercept
indicates where line crosses y-axis any curve will always cross y-axis when x=0
so to find y-intercept plug in 0 for x find y
y=4-x
rearrange= y=-1x+4
slope=-1 so down 1
and y intercept is (0,4)
y=2x-4
slope = think of this as a slope of 2= 2/1, so up 2 over 1 to the right
y-intercept is -4
so points are (0,-4), (1,-2), (2,0)
the area of a parallelogram
remember bxh but b and h have to be perpendicular so make sure you drop down create a perpendicular line for the height!
parallelogram triangles
when you split any parallleogram by its diagnol, you create two identical triangles. In this case, since triangle ABC has an area of 12, triagnle ACD must also have an area of 12!!!!
Does the point (-3,5) lie on the line y=-2x-1?
plug in -3 for x, into the equation, if hte math works and hte equation equals 5 then this point does lie on this line. If the math does not work , then this point does not lie on this line!
5=6-1
5=5 TRUE the point does lie on the line
what is equal to each other
circle equations!!!!
inscribed arc….
all this is saying is that if this angle is x then inside is 2x
Arc AOB has a measure of 30 degrees is the same as saying this is equal to 30 degrees, equal to measure of inscribed angle
if this is 60 then the inscribed angle has arc of 60
exterior angles of a triangle
- Exterior angles= sum of the two remote interior angles
- For a exterior angle is x, for b the exterior angle is y, c exterior angle is z, so a+b+c=180, x+y+z=360
- Each interior angle only has one exterior angle for ex x is the only exterior angle to a, only referring to each interior angle having one exterior angle can draw on either side
exterior angles for a hexagon.
if tok all those exterior angles for a hexagon it would equal 360
so x .. etc equal 360, interior angles = 540
