Unit 7 Triangles and Area

- This unit begins to classify triangles.
- It addresses special right triangles and the

Pythagorean Theorem (again). - This unit covers the area equations required for

quadrilaterals, and for circles. - It also covers the area of any polygon using an

apothem. - This unit differentiates between perimeter and

area similarity ratios, and concludes with

Geometric Probability.

Standards

- SPIs taught in Unit 7
- SPI 3108.1.1 Give precise mathematical

descriptions or definitions of geometric shapes

in the plane and space. - SPI 3108.1.2 Determine areas of planar figures by

decomposing them into simpler figures without a

grid. - SPI 3108.4.3 Identify, describe and/or apply the

relationships and theorems involving different

types of triangles, quadrilaterals and other

polygons. - SPI 3108.4.6 Use various area of triangle

formulas to solve contextual problems (e.g.,

Herons formula, the area formula for an

equilateral triangle and A ½ ab sin C). - SPI 3108.4.7 Compute the area and/or perimeter of

triangles, quadrilaterals and other polygons when

one or more additional steps are required (e.g.

find missing dimensions given area or perimeter

of the figure, using trigonometry). - SPI 3108.4.11 Use basic theorems about similar

and congruent triangles to solve problems. - SPI 3108.4.12 Solve problems involving

congruence, similarity, proportional reasoning

and/or scale factor of two similar figures or

solids. - SPI 3108.5.1 Use area to solve problems involving

geometric probability (e.g. dartboard problem,

shaded sector of a circle, shaded region of a

geometric figure). - CLE (Course Level Expectations) found in Unit 7
- CLE3108.2.3 Establish an ability to estimate,

select appropriate units, evaluate accuracy of

calculations and approximate error in measurement

in geometric settings. - CLE 3108.3.1 Use analytic geometry tools to

explore geometric problems involving parallel and

perpendicular lines, circles, and special points

of polygons. - CLE 3108.4.6 Generate formulas for perimeter,

area, and volume, including their use,

dimensional analysis, and applications. - CLE 3108.4.8 Establish processes for determining

congruence and similarity of figures, especially

as related to scale factor, contextual

applications, and transformations. - CLE 3108.5.1 Analyze, interpret, employ and

construct accurate statistical graphs. - CLE 3108.5.2 Develop the basic principles of

geometric probability.

Standards

- CFU (Checks for Understanding) applied to Unit 7
- 3108.1.5 Use technology, hands-on activities, and

manipulatives to develop the language and the

concepts of geometry, including specialized

vocabulary (e.g. graphing calculators,

interactive geometry software such as Geometers

Sketchpad and Cabri, algebra tiles, pattern

blocks, tessellation tiles, MIRAs, mirrors,

spinners, geoboards, conic section models, volume

demonstration kits, Polyhedrons, measurement

tools, compasses, PentaBlocks, pentominoes,

cubes, tangrams). - 3108.4.9 Classify triangles, quadrilaterals, and

polygons (regular, non-regular, convex and

concave) using their properties. - 3108.4.10 Identify and apply properties and

relationships of special figures (e.g., isosceles

and equilateral triangles, family of

quadrilaterals, polygons, and solids). - 3108.4.11 Use the triangle inequality theorems

(e.g., Exterior Angle Inequality Theorem, Hinge

Theorem, SSS Inequality Theorem, Triangle

Inequality Theorem) to solve problems. - 3108.4.12 Apply the Angle Sum Theorem for

polygons to find interior and exterior angle

measures given the number of sides, to find the

number of sides given angle measures, and to

solve contextual problems. - 3108.4.20 Prove key basic theorems in geometry

(i.e., Pythagorean Theorem, the sum of the angles

of a triangle is 180 degrees, characteristics of

quadrilaterals, and the line joining the

midpoints of two sides of a triangle is parallel

to the third side and half its length). - 3108.4.28 Derive and use the formulas for the

area and perimeter of a regular polygon.

(A1/2ap) - 3108.4.43 Apply the Pythagorean Theorem and its

converse to triangles to solve mathematical and

contextual problems in two- or three-dimensional

situations. - 3108.4.44 Identify and use Pythagorean triples in

right triangles to find lengths of an unknown

side in two- or three-dimensional situations. - 3108.4.45 Use the converse of the Pythagorean

Theorem to classify a triangle by its angles

(right, acute, or obtuse). - 3108.4.46 Apply properties of 30 - 60 - 90 and

45 - 45 - 90 to determine side lengths of

triangles. - 3108.5.2 Translate from one representation of

data to another (e.g., bar graph to pie graph,

pie graph to bar graph, table to pie graph, pie

graph to chart) accurately using the area of a

sector. - 3108.5.3 Estimate or calculate simple geometric

probabilities (e.g., number line, area model,

using length, circles).

Pythagorean Theorem

- In a RIGHT triangle (a triangle with one 90

degree angle), the sum of the squares of the

lengths of the legs is equal to the square of the

length of the hypotenuse (the longest side). - a2 b2 c2

a

c

b

Pythagorean Triple

- A set of non-zero whole numbers a, b, and c that

satisfy the equation a2 b2 c2 - Some triples include
- 3,4,55,12,138,15,177,24,25
- Recognize 3,4,5 and 5,12,13 they are the most

commonly used triples on standardized tests! - If you multiply each number in a Pythagorean

triple by the same whole number, the 3 numbers

that result also form a Pythagorean triple. - For example 3,4,5 x the whole number 2 equals 6

(3 x 2), 8 (4 x 2), and 10 (5 x 2), a new triple

Converse of the Pythagorean Theorem

- If the square of the length of one side of a

triangle is equal to the sum of the squares of

the lengths of the other 2 sides, then the

triangle is a right triangle. - In other words, if you calculate a2 b2 and it

does in fact c2, then in fact you can conclude

that it is a right triangle

Proofs of the Pythagorean Theorem

- The Brides Chair
- The area of a square is Side x Side (or side

squared) - Here, the side is a b. So the area of the

square is (ab)2 - The area of a triangle is ½ b x h
- This makes sense because the area of a rectangle

is base times height, so the area of a triangle

would be half of that - Continuing on then, the area of the triangle in

the square then is ½ a x b - There are 4 of these triangles, so the area would

be 4 x ½ x a x b - Adding the area of the center square (c2), we get

(ab)2 2ab c2, which simplifies to

a2 b2 c2

A Proof Discovered by a High School Student

- There are easily over 400 proofs of the

Pythagorean Theorem - This one was discovered by a high school student

(Jamie deLemos) in 1995.

- We will learn that the area of a trapezoid is the

top base plus the bottom base divided by 2 (in

other words we average the bases) x the height. - Here, that would be (2a 2b)/2 x ab
- The area of a triangle is ½ b x h
- This would be 2ab/2 2ba/2 2c²/2 for all of

the triangles - If you set them equal to each other, and reduce,

you get a2 b2 c2 - Remember, all of this is used to prove the

LENGTHS of the sides, not area

Area of a Triangle

- The area of a triangle is half the product

(multiply) of a base and the corresponding height - So, A 1/2 x b x h
- The base of a triangle can be any of its sides.

The corresponding height is the length of the

altitude to the line containing that base. - Right triangles are unique, in that you can pick

a base that automatically has a corresponding

height

h

b

Find the area of an Isosceles Triangle

- Here, the height is not so easily seen as it is

in a right triangle. To find the height, we draw

a line perpendicularly from the base to the

highest point of the triangle - To calculate the height, we can project a Right

Triangle, and determine the length of the 3rd

side. - 102h2 122
- h2 122-102
- h 6.6
- Therefore, the area of this triangle is ½ x 6.6 x

20 - Or, area 66 m2

12 m

h

20 m

Obtuse Triangles

- If the square of the length of the longest side

of a triangle is greater than the sum of the

squares of the length of the other 2 sides, the

triangle is obtuse - If C2gtA2B2 then the triangle is obtuse

A

B

C

Acute Triangles

- If the square of the length of the longest side

of a triangle is less than the sum of the squares

of the lengths of the other 2 sides, the triangle

is acute. - If C2ltA2B2 then the triangle is acute

B

A

C

Assignment

- Page 495-96 7-22,24-32
- Page 497 36-42
- Worksheet Practice 8-1
- Worksheet 7-1

Unit 7 Quiz 1

- Find the missing variable
- A 6, B 9, C ?
- A 4, B ? C 11
- A ? B 4, C 13
- A 2, B 13, C ?
- A 7, B ? C 14
- D 12, L 18, H ?
- E 11, D 11, L 20, H ?
- H 14, D 20, L ?
- L 6, H 4, D ?
- D 22, L 30, H ?

- A

C - B
- D E
- H
- L

From "A Few Good Men"

- "Son, we live in a world that has walls. And

those walls have to be guarded by men with guns.

Who's gonna do it? You? You, Lt. Weinberg? I have

a greater responsibility than you can possibly

fathom. You weep for Santiago and you curse the

Marines. You have that luxury. You have the

luxury of not knowing what I know that

Santiago's death, while tragic, probably saved

lives. And my existence, while grotesque and

incomprehensible to you, saves lives...You don't

want the truth. Because deep down, in places you

don't talk about at parties, you want me on that

wall. You need me on that wall. We use words like

honor, code, loyalty...we use these words as the

backbone to a life spent defending something. You

use 'em as a punchline. I have neither the time

nor the inclination to explain myself to a man

who rises and sleeps under the blanket of the

very freedom I provide, then questions the manner

in which I provide it! I'd rather you just said

thank you and went on your way. Otherwise, I

suggest you pick up a weapon and stand a post.

Either way, I don't give a dang what you think

you're entitled to!"

45-45-90 Triangle

- There are two Special right Triangles
- The first is a 45-45-90 Right Triangle
- If you take a square, and draw a diagonal line

through it, you get a 45-45-90 Right Triangle. - The acute angles of an Isosceles Right triangle

are both 45 degree angles because you are

bisecting a 90 degree angle - If each leg has length X (because it is a square,

the sides are even), and the hypotenuse has

length C, you can solve for C - s2s2C2 Pythagorean theorem
- 2s2C2 Simplify
- v2sC Take the square root of each side
- C sv2 Rewrite C in terms of X

45

C

s

90

45

s

45-45-90 Triangle Theorem

- In a 45-45-90 triangle, both legs are congruent

and the length of the hypotenuse is v2 times the

length of a leg, or - Hypotenuse sv2
- Side (Hv2)/2
- If S 6 on this right triangle,
- What is the hypotenuse?

s

2

90

45

45

sv2

Example

Remember, h sv2

- Assume these triangles are 45-45-90 triangles
- What is the value of each variable

y 6v2,

6

2v2

9

y

h 9v2

h

x

x (2v2)(v2) 2x2 4

Real World

- A high school softball diamond is a square. The

distance from base to base is 60 feet. How far

does a catcher throw the ball from home plate to

second base?

2nd Base

60 ft

60 ft

60 ft

60 ft

Home Plate

Solution

- The distance from home plate to 2nd base (d) is

the length of the hypotenuse of a 45-45-90

triangle - d sv2
- 60v2 so d84.85
- Or about 85 ft.

450

2nd Base

d

60 ft

900

Home Plate

Assignment

- Page 503 7-12 (keep this we will add to it)

Unit 7 Quiz 2

H

S

Given Triangle 1 is a 45/45/90

S

- Write the equation used to solve for the

hypotenuse (h) in a 45/45 right triangle - If S 6 in triangle 1, what is H?
- If S 5 in triangle 1, what is H?
- If S v2, in triangle 1, what is H?
- If H 4, in triangle 1, what is S?
- Write the equation used to solve for the side (s)

in a 45/45 right triangle - If S 5 in triangle 1, what is the area of the

triangle? - If S 8 in triangle 1, what is the area of the

triangle? - If H 4 in triangle 1, what is the area of the

triangle? - If H 6v2 in triangle 1, what is the area of the

triangle?

30-60-90 Triangle Theorem

- Another type of special right triangle is the

30o- 60o- 90o triangle - The length of the hypotenuse is twice the length

of the shorter leg - The length of the longer leg is v3 times the

length of the shorter leg. - Hypotenuse 2s
- Longer leg sv3
- Shorter leg is H/2 or (Lv3)/3
- NOTE The short leg is key to both

H2s

600

s and- sH/2 -and- s(Lv3)/3

300

Lsv3

Example

- Find the value of each variable
- Hypotenuse 2 times the shorter leg
- Or s H/2
- 82x, or x 4
- Longer leg v3 x shorter leg
- So y x v3 , or y 4 v3

60

8

x

30

y

Example

- Solve for d and f
- Long side v3 x Short
- 5 v3 x d
- d 5/ v3 multiply top and bottom by v3
- d (5v3)/3
- Hypotenuse 2 x short side
- f 2 x (5v3)/3
- f 10v3/3

60

f

d

30

90

5

Assignment

- Page 504 15-20, 23-28
- Worksheet Practice 8-2
- Worksheet 7-3

Create your own quiz

- Pair up with ONE person
- Create a 10 question quiz using 5 45-45 and 5

30-60 right triangles - Draw a triangle for each question
- Label one side, and label the other sides with a

variable (such as X, Y etc.) - Create an answer key
- You have 15 minutes
- Someone WILL be taking YOUR quiz!

Unit 7 Quiz 3

- The letter H stands for what (in both

triangles)? - The letter S stands for what in the 30-60-90

triangle? - The letter L stands for what in the 30-60-90

triangle? - If S4 in triangle 1, what is the value of H?
- If S6 in triangle 1, what is the value of H?
- If S 6 in triangle 1, what is the area?
- If S4 in triangle 2, what is the value of H?
- If S3 in triangle 2, what is the value of L?
- If H8 in triangle 2, what is the value of S?
- What is the formula for the area of a triangle?

- NOTE Triangle 1 is a 45-45-90 right triangle
- NOTE Triangle 2 is a 30-60-90 right triangle

H

H

S

S

S

L

Area of a Parallelogram

- The area of a parallelogram is the base times the

height (b x h) - Here is why ?
- Remember the area of a rectangle is base times

height also

Basically you can cut a parallelogram in half,

and put the two halves next to each other to make

a rectangle

Height

Base

Perimeter

- The perimeter of a polygon is the sum of the

lengths of the sides it is the measure of how

far around it is like the perimeter of your back

yard, you would measure how far around your back

yard it is.

Perimeter

- For any regular polygon, having

n-sides, the perimeter is n x the length of one

side (length s) - So a square would be 4s, a pentagon would be 5s

and so on - Otherwise, just add the lengths of the sides for

a given irregular polygon

Area

- For simple polygons which have 90 degree angles

like squares and rectangles, the area is equal

to side x side - For a square, we write this S x S or S2
- For a rectangle we write this L x W

S

W

S

L

Example

- What if you have an irregular shape like below?
- Pick out different shapes and add them up
- 12 cm 8 cm 4 cm 24 cm2

6 cm

this is 6 cm x 2 cm, for a total area of 12 cm

This is (22) cm x 2 cm, for a total of 8cm

This is 2 x 2 for 4 cm

2 cm

Area of a Trapezoid

- The area of a trapezoid is ½ times the height

times the top base (b1) the bottom base (b2) - In other words, you average the top and bottom

(add them and divide by 2) and multiply by the

height

Area of a Trapezoid

- Here is how the area of a trapezoid (1/2h(b1b2)

works ? - Take a trapezoid, and make a mirror image
- Rotate the mirror image
- Now we have a parallelogram again, with the same

height, but the base is b1b2 So the area of our

new parallelogram is base times height, or

(b1b2)xh. Since we only need one of them (we

only need 1 trapezoid, not 2), we use 1/2h(b1b2)

B1B2

Example

- To find the area of the trapezoid calculate

1/2h(b1b2) - (b1b2) 5 7 12
- We dont know the height
- We do have a 30/60/90 right
- triangle.
- We know the short side is 2 (7-52)
- So the long side (which is also the height) is

2v3 - Therefore the area is ½(2 v3)(57) 12 v3

5 m

2v3

600

7 m

2

Area of a Rhombus or a Kite

- The area of a Rhombus or a Kite is ½ x D1 x D2

(diagonal 1 and diagonal 2) - Lets say D1 6 and D2 6
- The area would be ½ x 6 x 6, or 18
- Imagine 1 triangle
- The area of this triangle is ½ x D1/2 x D2/2
- This would be ½ x 6/2 x 6/2 or ½ x 3 x 3 4.5
- Since there are 4 triangles, we would multiply 4

x 4.5 18 - The area of a kite can be proved similarly

D1

D2

Example

- Find the area of this kite
- A ½ x D1 x D2
- A ½ x (33) x (52)
- A ½ x 6 x 7
- A ½ x 42
- A 21

2

3

3

5

Example

15

12

- Find the area of this Rhombus
- A ½ x D1 x D2
- We know that D1 12 x 2 24
- We need D2
- We can either solve using the Pythagorean

theorem, or we can recognize the Pythagorean

Triple (3,4,5) - So the third side of the triangle is 9 (3 x 3, 4

x 3, and 5 x 3) - Therefore D2 9 x 2 18
- A ½ x 24 x 18
- A 216

9

Assignment

- Page 619 8-16
- Page 626 11-25
- Worksheet 7-1
- Worksheet 7-4

Unit 7 Quiz 2

- What is the area equation for a triangle?
- What is the area equation for a parallelogram?
- What is the area equation for a rectangle?
- What is the area equation for a square?
- What is the area equation for a Trapezoid?
- What is the area equation for a Kite?
- What is the area equation for a Rhombus?
- When C2gtA2B2 the triangle is _______
- When C2A2B2 the triangle is _______
- When C2ltA2B2 the triangle is _______

Unit 7 Quiz 4 You must draw a picture for each

problem, and show the equation used to solve each

question

- Tom cuts a square table in half diagonally, and

measures the diagonal, which is 7 feet long. - How long is each side of the table (to the

nearest foot)? - What is the area of the half of the table that

Tom kept, so that he can buy some paint (to the

nearest square foot)? - Jesse makes a bike ramp. He wants the ramp to be

20 feet long across the ground. He wants the ramp

to have a 30 degree angle rising to the top. - How high is the ramp at the other end?
- How long will the top of the ramp be?
- Nick needs to cut a cable to hook to the top of a

15 foot flag pole with the other end staked to

the ground. He wants the cable to be at a 45

degree angle. - How long is Nicks cable?

A look at the Circle

Diameter

Perimeter

.

Radius

Chord

Center Point

Circles

- A Circle is a set of points in a plane that are

the same distance from the center point - A Radius is a segment of a line which has one

endpoint on the center point, and the other

endpoint on the circle - A Diameter is a segment (or chord) that passes

through the center of the circle, and has an

endpoint on each side of the circle - A Chord is any segment whose endpoints are on the

circle - The Circumference is the distance around the

circle. - To calculate circumference, you multiply the

Diameter times pi, or 2 Radii times pi

Circumference

- Remember, circumference is the distance around

the circle - The ratio of the circumference to the diameter of

the circle is represented as - CDp or C 2pR (1 diameter 2 radii)
- This is called Diameter Pi or 2 Pi R
- p is the symbol for pi, which is an often used

number ? 3.14159. Pi goes on forever, but your

calculator can give you a good approximation

Find the Circumference

- A circle has a diameter of 10 feet. How far

around is it to the nearest tenth of a foot? - C D x Pi
- C 10 x 3.14
- C 31.4 FT

10 Feet

Check on Learning

- Find the circumference for the following circles
- A circle with a diameter of 2 4/5 inches
- 8.79 inches
- A circle with a radius of 30 mm
- 188.495 mm
- A circle with a diameter of 200 miles
- 628.318 miles
- A circle with a radius of 14 feet
- 87.964 feet

Area of a Circle

- The area of a circle is calculated as
- pR2
- This is called pi R squared
- Note We can take the Diameter (D) and divide it

in two to get the Radius (R) - We have to do this operation BEFORE we square it
- So we could write this p(D/2)2

More about p

- Its been calculated for thousands of years

Culture/Person Approximate Time Value Used

Babylonians 2000 BC 3 1/8 3.125

Egyptians 2000 BC 3.16045

China 1200 BC 3

Bible mentions it 550 BC 3

Archimedes 250 BC 3.1418

Hon Han Shu 130 sqrt (10) 3.1622

Ptolemy 150 3.14166

Bible Ref He made the Sea of cast metal, circular in shape, measuring ten cubits from rim to rim and five cubits high. It took a circumference of thirty cubits to measure around it. Bible Ref He made the Sea of cast metal, circular in shape, measuring ten cubits from rim to rim and five cubits high. It took a circumference of thirty cubits to measure around it. Bible Ref He made the Sea of cast metal, circular in shape, measuring ten cubits from rim to rim and five cubits high. It took a circumference of thirty cubits to measure around it.

More about p

- William Jones, a self-taught English

mathematician born in Wales, is the one who

selected the Greek letter for the ratio of a

circle's circumference to its diameter in 1706. - is an irrational number. That means that it can

not written as the ratio of two integer numbers.

For example, the ratio 22/7 is a popular one used

for but it is only an approximation which equals

about 3.142857143...Another more precise ratio

is 355/113 which results in 3.14159292... - Another characteristic of as an irrational number

is the fact that it takes an infinite number of

digits to give its exact value, i.e. you can

never get to the end of it. - Since 4,000 years ago and up until this very day,

people have been trying to get more and more

accurate values for pi. Presently supercomputers

are used to find the value of with as many digits

as possible. Pi has been calculated with a

precision containing more than one billion

digits, i.e., more that 1,000,000,000 digits!

More about p

- Egyptologists and followers of mysticism have

been fascinated for centuries by the fact that

the Great Pyramid at Gaza seems to approximate

pi. The vertical height of the pyramid has the

same relationship to the perimeter of its base as

the radius of a circle as to its circumference. - The first 144 digits of pi add up to 666 (which

many scholars say is the mark of the Beast).

And 144 (66) x (66). - If the circumference of the earth were calculated

using p rounded to only the ninth decimal place,

an error of no more than one quarter of an inch

in 25,000 miles would result. - In 1995, Hiroyoki Gotu memorized 42,195 places of

pi and is considered the current pi champion.

Some scholars speculate that Japanese is better

suited than other languages for memorizing

sequences of numbers.

Assignment

- Page 64 10-13, 23-33
- Worksheet 1-7
- Circles Worksheet
- Discovering Pi

Unit 7 Quiz 3

- A kite has a diagonal of 12 and a diagonal of 11.

What is its area? - A rhombus has a diagonal of 2 and a diagonal of

8. What is its area? - A square has a side 20 inches. What is its

area? - A rectangle has a base of 16, and a height of 5.

What is its area? - A parallelogram has a base of 8 and a height of

8. What is its area? - A triangle has a base of 18 and a height of 10.

What is its area? - A triangle has a base of 11 and a height of 8.

What is its area? - A trapezoid has a top base of 8, a bottom base of

9, and a height of 6. what is its area? - A trapezoid has a top base of 4, a bottom base of

20, and a height of 23. What is its area? - A parallelogram has a base of 20 and a height of

23. What is its area?

Unit 7 Quiz 5

- Match the equation with the definition

- Equation

- Area of a Triangle
- Area of a Parallelogram or Rectangle
- Area of a Rhombus or Kite
- Area of a Trapezoid
- Area of a Square
- Length of a side on a 45-45 triangle
- Length of the hypotenuse on a 30-60-90 triangle
- Length of the hypotenuse on a 45-45 triangle
- Length of the short side of a 30-60-90 triangle
- Length of the long side of a 30-60-90 triangle

- S (Hv2)/2
- A S2
- A (B x H)/2
- L Sv3
- A H x (B1 B2)/2
- S H/2 or S Lv3/3
- A B x H
- H 2S
- A (D1 x D2)/2
- H Sv2

Unit 7 Quiz 4 ?10 Points

- How do you calculate pi?
- Explain in your own words

Apothem

- The apothem of a regular polygon is a line

segment from the center to the midpoint of one of

its sides-NOT THE CORNER. - This line must be perpendicular to the side
- Regular polygons are the only polygons that have

apothems. - Because of this, all the apothems in a given

polygon will be congruent and have the same

length.

Center of Regular Polygon

Apothem

Area of a Regular Polygon

- The apothem can be used to find the area of any

regular n-sided polygon of side length s

according to the following formula - A (nsa)/2 or pa/2
- Here Area the number of sides (n) times the

length of one side (s) (which is the total

perimeter) times the apothem (a) all divided by 2 - Or, Area the perimeter (p) times the apothem

(a) all divided by two

Example

- Find the area of this regular decagon with a 12.3

inch apothem and 8 inch sides - A (n x s x a)/2
- A (10 x 8 x 12.3)/2
- A 492 inches2
- Find the area of a regular pentagon with 11.6 cm

sides and an 8 cm apothem

12.3

8

A (n x s x a)/2 A (5 x 11.6 x 8)/2 A 232 cm2

Example

- Find the area of this regular hexagon
- A (n x s x a)/2
- A (6 x 10 x a)/2
- Need a.
- Since we have a 30/60 triangle,
- a 5v3
- A (6 x 10 x 5v3)/2 259.80 mm2

600

10mm

300

a

5mm

Example

18

- Find the area of this regular triangle
- A (n x s x a)/2
- A (3 x 18 x a)/2
- We need the apothem
- This triangle is a 30/60/90
- The apothem a is the short side of the triangle
- The short side is calculated as Lv3/3
- Or 9v3/3
- So the Apothem is 5.19
- A (3 x 18 x 5.19)/2
- A 140.13

a

10.39

Assignment

- Page 672/673 16-24

Unit 7 Quiz 6

- Write the equation to calculate p

Similarity

- Polygons are similar if they have the exact same

measure of degree for all angles, and that all

sides are proportional they have the same ratio - The similarity ratio is found by writing the

length of one side over the length of the same,

corresponding side from the second polygon - For example, the similarity ratio here is 2/3

(3/4.5 2/3)

3 4.5

2 3

2 3

4 6

Perimeters and Area of Similar Polygons

- If you have a similarity ratio between two

polygons ? for example a/b, then - The similarity ratio for the perimeter is the

same ? that is a/b - The similarity ratio for the area is a little

different ? it is a2/b2 - We can remember this because when we label area

it is always squared for example cm2

Example

- These polygons are similar
- What is the similarity ratio?
- 6/9, which reduces to 2/3
- If the perimeter of the small polygon is 20 m,

what is the perimeter of the large polygon? - 2/3 20/P, 2P 60, P 30
- If the area of the small polygon is 60 m2, what

is the area of the large polygon? - 22/32 60/A, 4/9 60/A,
- Cross multiply and divide 4A 60x9,
- A 135

6 m

9 m

Example

- The area of the small pentagon is about 27.5 cm2
- What is the area, A, of the large pentagon?
- 4/10 2/5
- 22/52 27.5/A
- 4/25 27.5/A
- 4A 25(27.5)
- 4A 687.5
- A 171.875 cm2

4 cm

10 cm

Notice, that you need to reduce the ratio (4/10

2/5) before you square the top and bottom, or you

will get the wrong answer

Find Similarity Ratios

- The area of two similar triangles are 50 cm2 and

98 cm2 - Find the similarity ratio A/B
- A2/B2 50/98 now simplify
- A2/B2 25/49 now square root all
- A/B 5/7
- This is the similarity ratio, and the perimeter

ratio

Assignment

- Page 638 9-16,19-29
- Worksheet 8-6

Unit 7 Quiz 5 All answers to the nearest 10th

- A circle has a radius of 5.
- What is its circumference?
- What is its area?
- A circle has a radius of 8.
- What is its circumference?
- What is its area?
- A circle has a diameter of 12.
- What is its circumference?
- What is its area?
- A circle has a diameter of 30.
- What is its circumference?
- What is its area?
- A circle has a circumference of 12p
- What is its diameter?
- What is its area?

A Toss of a Coin

I inch Radius

- You are at a carnival, where they have a coin

toss - They have an 8 inch square, with a one inch

radius circle on the square - If you toss a quarter and the entire quarter is

on the circle, you win a prize - What is the probability of winning?

8 inches

The Solution

- A quarter is about 1 inch in diameter
- Therefore, to be completely on the 2 inch circle,

the quarter has to be at least ½ in from the edge

of the circle - This means the desired area is a circle 1 inch in

diameter (2 inch total, minus ½ inch on each

side, 1 inch) - The total area is the 8 inch square
- To calculate Probability, we calculate the ratio

of the desired outcome divided by the total

outcome - Therefore p(.50)2/82 .012 or 1.2
- Not very good ODDS!!

8 inches

Probability

- The probability of an event occurring is the

ratio of the number of favorable outcomes to the

number of possible outcomes - A geometric probability model is one in which we

use points to represent outcomes. (We will also

use area). You find probabilities by comparing

measurements of sets of points. For example if

points of segments (like a number line) represent

possible outcomes, then ? - P(event) length of favorable segment
- length of entire segment

Example

- Suppose a fly lands on a 12 inch rulers edge.

What is the probability that the fly lands on a

point between 3 and 7? - P(landing between 3 and 7)
- Length of favorable segment 4 or 1
- Length of total segment 12 3

1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 - 9

- 10 - 11 - 12

4

12

Another example

- A point on segment AB is selected at random. What

is the probability that it is a point found on

segment CD? - How many favorable outcomes are there?
- Or, what is the length of the favorable segment?
- 4
- How many possible outcomes are there?
- Or, what is the length of the entire segment?
- 10
- P(point on CD) length CD/length AB or 4/10 or

2/5

A C

D B

0 1 2 3 4 5 6

7 8 9 10

Bus example

- Elenas bus runs every 25 minutes.
- If she arrives at her stop at a random time, what

is the probability that she will have to wait at

least 10 minutes for the bus? - Assume the stops take very little time

Bus Example

There are 2 ways to solve this One way is the

same as we did for the first problem 10/25

2/5 The other way is this The total possible

probabilities will add up to one. Here, we have

2 probabilities that must add up to one. The

first is 3/5, the other is 2/5 for a total of

5/5

- Let segment AB represent the 25 minute wait

between buses. - Let segment AC represent any random wait, up to

10 minutes before the bus arrives - Therefore if Elena arrives anywhere on segment

AC, she will have to wait at least 10 minutes - P(waiting at least 10 min) length AC/length AB
- Or 15/25 or 3/5
- What is the probability that Elena will have to

wait less than 10 minutes?

?

A C

B

0 5 10 15

20 25

Probability using regions or Area

- If the points of a region represent equally

likely outcomes, then you can find probabilities

by comparing areas - Where P(event) area of favorable region/ area

of entire region

Target example

- Assume that a dart you throw will land on a 1

foot square dartboard and is equally likely to

land at any point on the board. Find the

probability of hitting each of the blue, yellow

and red regions. The radii of the concentric

circles are 1, 2 and 3 inches respectively.

Target Example

- Radii 1 in (Blue) 2 in (Yellow) and 3 in (Red)
- P(Dart in blue) area of blue/total area
- p(1)2/122
- which equals p/144 or .022, or 2.2
- P(Dart in yellow) area of yellow/total area
- (p(2)2 - p(12))/144
- Which equals 3p/144 or .065 or 6.5
- P(Dart in red) area of red/total area
- (p(3)2 - p(22))/144
- Which equals 5p/144 or .109 or 10.9
- What if the radius of the blue circle was

doubled? What would the probability be of hitting

the blue circle? - What if it was tripled? What would be the

probability of hitting the blue circle?

12

12

4p/144 8.73

9p/144 19.63

Assignment

- Page 671 8-16
- Page 672 18-26
- Bulls-eye Worksheet
- Worksheet 9 Geometric Probability problems
- Worksheet 7-8
- Geometric Probability quiz

Unit 7 Quiz 6 All answers to the nearest 10th

- A circle has a radius of 9.
- What is its circumference?
- What is its area?
- A circle has a radius of 11.
- What is its circumference?
- What is its area?
- A circle has a diameter of 13.
- What is its circumference?
- What is its area?
- A circle has a diameter of 40.
- What is its circumference?
- What is its area?
- A circle has a circumference of 14p
- What is its diameter?
- What is its area?

Unit 7 Quiz 7 ?2 points each

Round all answers to whole numbers

- Find the area of a parallelogram with a base of 3

cm and a height of 4 cm - Find the area of a trapezoid with a top base of 3

cm, a bottom base of 5 cm, and a height of 5 cm - Find the area of a kite with a diagonal of 5 cm

and a second diagonal of 7 cm - Find the area of a circle with a radius of 5

meters - Find the circumference of a circle with a

diameter of 10 inches - Bonus for 2 points Write the equation to

calculate p

Unit 7 Quiz 7

Round to 2 decimal places

- Use the following information to calculate the

area of each slice of pie

Car Owner

Mustang 41

2. Corvette 10

3. Camaro 25

4. G8 8

5. BMW 5 5

6. Jetta 7

7. Viper 4

Radius 10 in.

To calculate a percentage of a total area,

convert the percentage to a decimal, and multiply

times the area

Unit 7 Quiz 7 (10 points)

- Jason built a kids playhouse for his daughter

Amy. - The playhouse is an exact replica of his house.
- The playhouse has a perimeter that is five times

smaller than the real house (15 or 1/5) - If the playhouse has an area of 100 square feet,

how many square feet is the real house?

Unit 7 Final Extra Credit

At a minimum, show equation and solution

Given You randomly throw a dart, calculate the

following

- 2 points each
- What is the probability of not hitting any

circle, but instead hitting the blue background

target? - What is the probability of hitting any ring red,

yellow or blue? - What is the probability of hitting only the red

ring? - What is the probability of hitting only the

yellow ring? - What is the probability of hitting the blue

circle?

12 inches

12 inches

The center circle has a diameter of 4 inches and

each succeeding circle has a 2 inch larger

diameter