Geometry

Question 1

arc length equations

Answer 1

central angle/360

sector area/circle area

arc length/circumference

Question 2

circumference of a circle

Answer 2

pi X d
or pi X 2r

Question 3

area of a circle

Answer 3

pi X r^2

Question 4

diameter of a circle

Answer 4

d=2r

Question 5

if central angle is 60 degrees

Answer 5

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

Question 6

secto area=

Answer 6

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

Question 7

arc length=

Answer 7

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

Question 8

a sector has a radius of 9 and an area of 27 pi. What is the central angle of the sector?

Answer 8

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

Question 9

sum of any two sides > third side

Answer 9

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

Question 10

Two sides of a triangle have lengths of 5 and 19. Can the third side have a length of 13?

Answer 10

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

Question 11

Two sides of a triangle have lengths 8 and 17. What is the range of possible values of the length of the third side?

Answer 11

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

Question 12

sum of three angles

Answer 12

the sum of internal angles of a traingle must be 180 degrees!!!!!

SUM of three angles = 180

Question 13

a straight line always has a measure of…..

Answer 13

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

Question 14

same side= same angle

Answer 14

-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

Question 15

isosceles

Answer 15

a triangle has two equal angles and two equal sides

Question 16

equilateral triangle

Answer 16

a triangle has three equal angles all 60 degrees and three equal sides

all 60 degrees!

Question 17

If you see two equal sides in a triangle

If you see two equal angles in a triangle

Answer 17

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

Question 18

the perimeter of a triangle

What is the perimeter of a triangle with sides 5, 8, and 12?

Answer 18

it is the sum of the lengths of all 3 sides!

5+6+10=21 perimeter

5+8+12 = 25 perimeter

Question 19

Area of a triangle

Answer 19

area of a triangle= 1/2 X base x Height
the base and the height must be perpendicular to each other

Question 20

pythagorean theorem

Answer 20

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

Question 21

quadrilaterals

Answer 21

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.

Question 22

parallelograms

Answer 22

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

Question 23

parallelograms 2

Answer 23

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

Question 24

rectangles

Answer 24

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

Question 25

square

Answer 25

square= rectangle+ four equal sides

Question 26

square 2

Answer 26

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

Question 27

x=-3

Answer 27

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

Question 28

y=1

Answer 28

every point on dotted line has a y coordiante of 1, horizontal line, these points form a horizontal line

Question 29

y=mx+b

y=3x-2
y=-x+4
y=1/2x

Answer 29

m=3, b=-2
m=-1, b=4
m=1/2, b=0

Question 30

m=

Answer 30

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

Question 31

b=

Answer 31

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

Question 32

y=4-x

Answer 32

rearrange= y=-1x+4

slope=-1 so down 1
and y intercept is (0,4)

Question 33

y=2x-4

Answer 33

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)

Question 34

the area of a parallelogram

Answer 34

remember bxh but b and h have to be perpendicular so make sure you drop down create a perpendicular line for the height!

Question 35

parallelogram triangles

Answer 35

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!!!!

Question 36

Does the point (-3,5) lie on the line y=-2x-1?

Answer 36

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

Question 37

what is equal to each other

Answer 37

Question 38

circle equations!!!!

Answer 38

Question 39

inscribed arc….

Answer 39

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

Question 40

exterior angles of a triangle

Answer 40

• 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

Question 41

exterior angles for a hexagon.

Answer 41

if tok all those exterior angles for a hexagon it would equal 360

so x .. etc equal 360, interior angles = 540