Geometry

Trapezoid

A quadrilateral with exactly one pair of parallel sides

supplementary angles

Two angles whose sum is 180 degrees

complementary angles

Two angles whose sum is 90 degrees

vertical angles

opposite angles formed by two intersecting lines that EQUAL.

There is a rectangular parking lot with a length of 2x and a width of x. What is the ratio of the perimeter of the parking lot to the area of the parking lot, in terms of x?

perimeter= 2(l+w) > 2(2x+x)= 6x Area= LW= (2x)(x) > 2x^2 3/x answer

A pentagon has three sides with length x, and two sides with the length 3x. If x is — of an inch, what is the perimeter of the pentagon?

perimeter= x+x+x+3x+3x=9x * 2/3 = 6

How to find the length of an arc

1. Find circumference 2. Use the radius 3. Equations 2pieR 4. use the central angle to determine what fraction the arc is of the circle. put it over 360 multiply the fraction by the circumference

How to find the area of a sector

1. Find the area 2.Then, use the central angle to determine what fraction of the entire circle is represented by the sector. 3. Multiply by area

Sum of interior angles

(n-2)180

sum of interior angles of a polygon

(n-2)180

Triangle A has a base of x and a height of 2x. Triangle B is similar to Triangle A, and has a base of 2x. What is the ratio of the area of Triangle A to Triangle B?

1/4

What is the length of arc AXB?

1/6(24pie)= 4pie

Randy can run n meters every 2 seconds. If the circular track has a radius of 75 meters, how many minutes does it take Randy to run twice around the track?

10 minutes: The distance around the track is the circumference of the circle: C = 2nr = 150piT Running twice around the circle would equal a distance of 300piT meters. If Randy can run n meters every 2 seconds, he runs 30piT meters every minute. Therefore, it will take him 10 minutes to run around the circular track twice

What is the area of an equilateral triangle whose sides measure 8 cm long?

16 square root 3 : Draw in the height of the triangle (see figure). If triangle ABC is an equilateral triangle, and ABD is a right triangle, then ABD is a 3 0 -6 0 - 90 triangle. Therefore, its sides are in the ratio of 1: -\/3 : 2. If the hypotenuse is 8, the short leg is 4, and the long leg is 4 V3 . This is the height of triangle ABC. Find the area of triangle ABC with the formula for area of a triangle: or use area of an equilateral triangle formula

Alexandra wants to pack away her posters without bending them. She rolls up the posters to put in a rectangular box that is 120 inches long, 90 inches wide, and 80 inches high. What is the longest a poster can be for Alexandra to pack it away without bending it (i.e., what is the diagonal of the rectangular box)?

170 inches: Using the Deluxe Pythagorean Theorem, calculate the length of the diagonal: 80^2+90^2+120^2= x^2

Surface area of a rectangular solid

2(height)(width) + (2(height)(depth)+ 2(width)depth

A square is bisected into two equal triangles (see figure). If the length of BD is 16 square root 2 inches, what is the area of the square?

256

Surface Area of a Cylinder

2πrh+2πr²

List a pythagorean triple for RIGHT Triangles

3 -4 -5 6 -8 -10 9-12-15 12-16-20 5-12-13

ABCD is a square picture frame (see figure). EFGH is a square inscribed within ABCD as a space for a picture. The area of EFGH (for the picture) is equal to the area of the picture frame (the area of ABCD minus the area of EFGH). If AB = 6, what is the length of EF1

3V2 : The area of the frame and the area of the picture sum to the total area of the image, which is 62, or 36. Therefore, the area of the frame and the picture are each equal to half of 36, or 18. Since EFGHis a square, the length of EFis yfl8 , or 3V2 .

BE and CD are both diameters of circle with center A (see figure). If the area of Circle A is 180 units2, what is the area of sector ABC + sector ADE1

40 units2: The two central angles, CAB and DAE, describe a total of 80°. Simplify the fraction to find out what fraction of the circle this represents: 80/360 > 2/9 > 2/9 times 180= 40

The points of a six-pointed star consist of six identical equilateral triangles, with each side 4 cm (see figure). What is the area of the entire star, including the center?

48 square root 3

What is the measure of angle x in the figure to the right?

50

Two sides of a triangle are 4 and 10. If the third side is an integer x, how many possible values are there forx?

7: If two sides of a triangle are 4 and 10, the third side must be greater than 10-4 and smaller than 10 + 4. Therefore, the possible values for x are {7, 8, 9, 10, 11, 12, and 13}. You can draw a sketch to convince yourself of this result:

The size of a square computer screen is measured by the length of its diagonal. How much bigger is the visible area of a square 24-inch screen than the area of a square 20-inch screen?

88

Area of a rhombus

A= Diagonal 1 + Diagonal 2/ 2

Area of a trapezoid

A=1/2(b1+b2)h

Area of a triangle

A=1/2bh

Area of any parallelogram

A=bh

Area of a rectangle

A=lw

Area of a circle

A=πr²

Solving for hypotenuse

ADD

Square

All angles are 90 degrees and all sides are equal

Square

All angles are 90 degrees and all sides are equal All the properties of rectangle, rhombus, and parallelogram

Rectangle

All angles are 90 degrees and opposite sides are equal

Rhombus

All sides are equal. Opposite angles are equal.

Given that an equilateral triangle has a side of length 10, what is its height?

Geometry Manhattan prep

In Triangle ABC, AD = DB = DC (see figure). Given that angle DCB is 60° and angle ACD is 20°, what is the measure of angle x?

If AD = DB = DC, then the three triangular regions in this figure are all isosceles triangles. Therefore, you can fill in some of the missing angle measurements as shown to the right. Since you know that there are 180° in the large triangle ABC, you can write the following equation: x + x

A cylinder has a surface area of 36077, and is 3 units tall. What is the diameter of the cylinder's circular base?

Use surface area equation plugin the values quadratic equation (r^2+3r-180=0) > r+15=0 r-12=0 R= -15 (get rid of this u dont need it) R= 12 Diameter= 2R so > 12(2)= 24

Big 4 properties of parallelograms

Opposite sides and opposite angles are equal; parallel opposite sides, diagonals bisect each other

Parallelogram

Opposite sides are equal opposite angles equal

Jane has to paint a cylindrical column that is 14 feet high and that has a circular base with a radius of 3 feet. If one bucket of paint will cover 1077 square feet, how many full buckets does Jane need to buy in order to paint the column, including the top and bottom?

SA equation ratio

Surface Area of a cube

SA=6s^2

40 percent of Andrea's living room floor is covered by a carpet that is 4 feet by 9 feet. What is the area of her living room floor?

The area of the carpet is equal to lx w, or 36 ft2. Set up a proportion to find the area of the whole living room floor: — = — Cross-multiply to solve. 100 x 40* = 3,600 * = 90 ft2

Given that the short leg of a 30-60-9 0 triangle has a length of 6, what are the lengths of the long leg and the hypotenuse?

The short leg, which is opposite the 30 degree angle, is 6. Use the ratio 1: y/3 :2 to determine that the multiplier x is 6. You then find that the sides of the triangle have lengths 6 :6 V3 :12. The long leg measures 6 y/3 and the hypotenuse measures 12.

A rectangular solid has a square base, with each side of the base measuring 4 meters. If the volume of the solid is 112 cubic meters, what is the surface area of the solid?

The volume of a rectangular solid equals (length) x (width) x (height). If you know that the length and width are both 4 meters long, you can substitute values into the formulas as shown: 112 = 4 x 4 x h h = 7 To find the surface area of a rectangular solid, sum the individual areas of all six faces: Area of one face Total area of two identical faces Top and Bottom: Sides: 4 x 4 = 1 6 4 x 7 = 28 -► 2 x 16 = 32 -► 4 x 2 8 = 112 All 6 faces 32 + 112 = 144 m2

A swimming pool has a length of 30 meters, a width of 10 meters, and an average depth of 2 meters. If a hose can fill the pool at a rate of 0.5 cubic meters per minute, how many hours will it take the hose to fill the pool?

The volume of the pool is (length) x (width) x (height), or 30 x 10 x 2 = 600 cubic meters. Use a standard work equation, R T = W, where W represents the total work of 600 m3: 0.5*= 600 t= 1,200 minutes Convert this time to hours by dividing by 60: 1,200 -5- 60 = 20 hours. Alternately, we could convert first: 0.5 m3/min x 60 min/hr = 30 m3/hr. Next, use the standard work equation: 30*= 600 t = 20 hours.

A rectangular tank needs to be coated with insulation. The tank has dimensions of 4 feet, 5 feet, and 2.5 feet. Each square foot of insulation costs $20. How much will it cost to cover the surface of the tank with insulation?

To find the surface area of a rectangular solid, sum the individual areas of all six faces: Area of one Total area of two face identical faces Top and Bottom: 5 x 4 = 20 —► 20 x 2 = 40 Side 1: 5 x 2.5 = 12.5 -► 12.5 x 2 = 25 Side 2: 4 x 2 .5 = 10 -► 1 0 x 2 = 20 All 6 faces -► 40 + 25 + 20 = 85 ft2 Covering the entire tank will cost 85 x $20 = $1,700.

Volume of a cube

V=s³

Volume of a cylinder

V=πr²h

Frank the Fencemaker needs to fence in a rectangular yard. He fences in the entire yard, except for one full side of the yard, which equals 40 feet. The yard has an area of 280 square feet. How many feet offence does Frank use?

You know that one side of the yard is 40 feet long; call this the length. You also know that the area of the yard is 280 square feet. In order to determine the perimeter, you must know the width of the yard. A — Ix w 280 = 40w w — 280 -5- 40 = 7 feet Frank fences in the two 7-foot sides and one of the 40-foot sides. 40 + 2(7) = 54

Rectangle properties

all angles are 90* and opposite sides are equal

Rhombus prop

all sides are equal. Opposite angles are equal

inscribed angle

an angle whose vertex is on the circle half of the central angle

Pythagorean Theorem

a²+b²=c²

Volume of a rectangular solid

height x depth x width

If the perimeter of a rectangular flower bed is 30 feet, and its area is 44 square feet, what is the length of each of its shorter sides?

look @ pdf explanation; answer is 4 tho

Solving for leg/side

subtract

Perimeter of a polygon

sum of the lengths of the sides

Perimeter of a polygon

sum of the lengths of the sides`

scale factor

the ratio of the lengths of two corresponding sides of two similar polygons

45-45-90 triangle

x, x, x√2

30-60-90 triangle

x, x√3, 2x