Welcome to our sixth seminar!

We’ll begin shortly

Line segments

A point is a spot in space that has no length, width or height. Think of it as a place holder.

A line segment is a line that ends at two points.For example:

A

B

This line segment is named AB or AB

Rays

Rays are lines with only one end point. You can think of them as starting at a point and then extending into infinity in one direction. For example:

C

D

This ray is called AB or AB

AnglesAn angle is made up of two rays, lines, or line segments

which start at the same endpoint (called the vertex). For example:

A

B

C R

S TJ K

L M

This angle is namedBے ,CBAے, ABCےor ے k

k m

a

This angle is named Sے ,TSRے ,RST ے or ے m

The highlighted angle is nameda ے LNJ, or ے ,JNLے

N

Types of angles

1800

900

3600

< 900> 900

Line segmentOne full rotation=3600

Straight angle = 1800

Right angle = 900

An acute angle is one that measures less than 900

An obtuse angle is one thatMeasures more than 900

More types of angles

Angles whose sum is 900 are called complementary angles.Examples:

Angles whose sum is 1800 arecalled supplementary angles

600 300

580

320 1340

460

600 +300 =900

320 +580 =900 460 +1340 =1800

Note that if you know one angle you can calculate the other:

380

The sum of a and 38 is 180a + 38 = 180 Subtract 38 from both sides:a = 180 – 38a = 1420

a

Planes (a 2 dimensional surface)

Pairs of linesParallel lines have the same distance between them at each point.They never intersect and the angle between them is 00

Perpendicular lines intersect at 900.

Intersecting lines meet at a point at an angle not 90 . The vertical angles (opposite) are equal to each other. The sumof the adjacent angles (next to each other) is 180 . Here Vertical angles are a,c, and b,d. Adjacent angles are a,b;b,c; c,d; and a,d.a

bc

d

a bc d

e f

g h

A transversal line is one that intersection two other lines.Corresponding angles are those on the same side of the Two lines. Here they are a,e; c,g; b,f; and d,hAlternate interior angles are those that are opposite Interior. Here they are c,f and d,e. If the two lines are parallel thenCorresponding lines are equal and alternate interior anglesAre also equal.

A few examples; solve for all angles

350

1220

a bc

d

e fh i

c and 35 are vertical angles and equal so c = 350

a and c are adjacent angles soa + c= 1800

a +35 = 1800

a = 1800 -350

a = 1450 and b = 1450 (opposite)Summary: a = 1450 , b = 1450 , c =350

1220 and i are corresponding angles; i = 1220

1220 and e; I and j are vertical angles; e=1220 , j=1220

d and 1220 are adjacent angles sod + 1220 = 1800

d = 1800 – 1220

d = 580

d and h are vertical angles so h = 580

d, f and h,k are vertical angles so f=580 and k = 580

Summary: d=580 , e=1220 , f=580 , h=580 , i=1220 , j=1220 ,k = 580

j k

Polygons (closed 2-D figures)

Types of triangles

Similar figures (same shape, different sizes)

A = A'

B = B'

C = C'

AB BC AC

A'B' B'C' A'C'

Example

Find ZX and ZY

ZX 12 =

5 3ZX

= 45

5ZX

5 = 4(5)

ZX = 20 units

ZY 12 =

4 3ZY

= 44

4ZY

4 = 4(4)

ZY = 16 units

Area and perimeter formulas

Perimeter The perimeter (P) of a polygon (a two-dimensional shape with at least

three sides) is the sum of all of the sides. In other words it is the distance around the shape. Don’t forget to include units; they will be in length (like m, ft, in, etc).

What is the perimeter of this trapezoid (a four sided shape with unequal sides)?

P = the sum of the sides

P = 28 + 15 + 8 + 12

P = 43 + 20

P = 63 units

NOTE: we ignored the extraneous information (the height). If there are no given units write “units”

A dollar bill has a width of 2.56in and a length of 6.14 inches. Find the perimeter.

The perimeter of a rectangle (which has two equal sides)

can be found by using the formula:

P = 2L + 2W Where L is length and W is width

Here W = 6.14in and L = 2.56in

P = 2(6.14) + 2(2.56) Multiply:

P = 12.28 + 5.12

P = 17.4 in

The perimeter of the dollar bill is 17.4 inches.

Example

If the radius of this circle is 4 inches,

what is the perimeter?

Use C = 2 r

C = 2(3.14)(4in)

C = 6.28(4in)

C = 25.13 in Round to the

nearest whole num

ber:

C = 25 in

A circle with a radius of 4 in has a perimeter of 25 in

This figure contains a rectangle which is 8 by 3 ft and a triangle with a height of 8 ft and a base of 6 – 3 or 3ft.

Add these two areas to get the total

r

2r

t

Step one: find the area of the rectangle

A = LW L = 3ft W = 8ft

A = (3ft)(8ft)

A = 24ft

Step two: find the area of the triangle:

1A = bh b = 3ft h = 8ft

21

A = 32

t

2t

total r t

2 2total

2total

2

ft (8ft) half of 8 is 4

A = 4ft(3ft)

A = 12ft

Step three: Add

A A + A

A = 24ft + 12ft

A = 36ft

The area of this figure is 36ft

Volume and formula problems

V=πr 2h

Surface area: the total amount of area on the surface of a three dimensional figure. The units are the same as area: length squared.

Here are some of the common formulas for finding surface area:

Rectangular solid:

S = the sum of all of the sides

For this diagram the surface area is:

S = 2lw + 2lh + 2wh

Or in this second picture add the areas:

S = 2A + 2B + 2C

Explaining the cylinder

Here is a picture of the surfaces of a cylinder that is enclosed. I has the area of each end and the rectangle that surrounds it.

If it was not enclosed or had only one end enclosed you remove those parts from the equation.

ExamplesFind the surface area of a closed

cylinder with a height of 16 inches and a diameter of 12 inches. The radius is half of 12. r = 6 in. round to the nearest tenth.

2

2

2 2

2 2

2 2

S = 2 r + 2 rh r = 6in h = 16in

S = 2(3.14)(6in) + 2(3.14)(6in)(16in) square and multiply:

S = 2(3.14)(36in ) 2(3.14)(96in )

S = 6.28(36in ) 6.28(96in )

S = 226in + 602.88in

S = 8

2

2

2

28.88in Round to the nearest tenth:

S = 828.9in

The surface area of the object is 828.9in

2

2

2

2

2

2

The surface are of a sphere is given by S = 4 r and r = 14cm

S = 4(3.14)(14cm)

S = 4(3.14)(196cm )

S = 12.47(196cm )

S = 2463cm

The surface area of this sphere is 2,463cm .

Volume: the amount of ‘stuff’ enclosed in a three dimensional object. Here is an example of what it looks

like. The units are length cubed.

3

This object has 72 "little unit

cubes inside of it because its

volume is 72units . We'll work

this in a moment.

3

3

3

3

3

3

4The volume of a sphere is given by: V = r

3 r = 14cm

4V = (3.14)(14cm)

34

V = (3.14)(2744cm )312.56

V = 2744cm3

34,464.64V = cm

3

V = 11,488.21cm

The volume of th

3is sphere is 11,488,21cm .

Examples

We found the surface area of this object, now let’s find the volume.

2

3

3

The volume of a rectangular solid is given by V = LWH

Let the units be cm's.

L = 8cm W = 3cm H = 4cm

V = (8cm)(3cm)(4cm)

V = (24cm )(4cm)

V = 96cm

The volume of this object is 96cm

Pyramid

This is a triangular pyramid that is 2000ft tall and each base length is 2500ft. What is the area?

2

1The volume of a triangular pyramid is V = lwh

3 l=2500ft w = 2500ft

h = 2000ft

1V = 2500ft 2000ft

31

V = 6,250,03

2

3

3

3

00ft 2000ft

1V = 12,500,000,000ft

3

V = 4,166,666,667ft

The volume of the pyramid is 4,166,666,667ft .

Euler’s formula:# of vertices - # edges + #face = 2

Example:If #vertices = 11, # faces = 5, find the # edges

# of vertices - # edges + #face = 2

11 - E + 2 = 213 - E = 2

13 - E – 13 = 2 – 13-E = -11E = 11

There are 11 edges

Converting square and cube units

2 2

22

2

Convert 15.6 ft to yd

1yd15.6 ft

12 ft

= 15.6 ft

2

2

1yd

144 ft

2

2 2

0.108 yd

15.6 ft is 0.108 yd

3 3

3

3

3

Convert 78.33 yd to ft

12 ft78.33 yd

1 yd

78.33 yd

3

3

1728 ft

1 yd

3

3 3

135354 ft

78.33 yd is 135,354 ft

Thank you for attending!