Unit 7 Triangles and Area

1
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.

2
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.

3
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).

4
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
5
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

6
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

7
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
8
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

9
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
10
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
11
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
12
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
13
Assignment

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

14
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

15
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!"

16
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
17
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
18
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
19
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
20
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
21
Assignment

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

22
Unit 7 Quiz 2
H
S
Given Triangle 1 is a 45/45/90
S

1. Write the equation used to solve for the
   hypotenuse (h) in a 45/45 right triangle
2. If S 6 in triangle 1, what is H?
3. If S 5 in triangle 1, what is H?
4. If S v2, in triangle 1, what is H?
5. If H 4, in triangle 1, what is S?
6. Write the equation used to solve for the side (s)
   in a 45/45 right triangle
7. If S 5 in triangle 1, what is the area of the
   triangle?
8. If S 8 in triangle 1, what is the area of the
   triangle?
9. If H 4 in triangle 1, what is the area of the
   triangle?
10. If H 6v2 in triangle 1, what is the area of the
    triangle?

23
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
24
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
25
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
26
Assignment

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

27
Create your own quiz

1. Pair up with ONE person
2. Create a 10 question quiz using 5 45-45 and 5
   30-60 right triangles
3. Draw a triangle for each question
4. Label one side, and label the other sides with a
   variable (such as X, Y etc.)
5. Create an answer key
6. You have 15 minutes
7. Someone WILL be taking YOUR quiz!

28
Unit 7 Quiz 3

1. The letter H stands for what (in both
   triangles)?
2. The letter S stands for what in the 30-60-90
   triangle?
3. The letter L stands for what in the 30-60-90
   triangle?
4. If S4 in triangle 1, what is the value of H?
5. If S6 in triangle 1, what is the value of H?
6. If S 6 in triangle 1, what is the area?
7. If S4 in triangle 2, what is the value of H?
8. If S3 in triangle 2, what is the value of L?
9. If H8 in triangle 2, what is the value of S?
10. 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
29
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
30
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.

31
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

32
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
33
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
34
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

35
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
36
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
37
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
38
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
39
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
40
Assignment

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

41
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 _______

42
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?

43
A look at the Circle
Diameter
Perimeter
.
Radius
Chord
Center Point
44
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

45
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

46
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
47
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

48
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

49
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.
50
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!

51
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.

52
Assignment

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

53
Unit 7 Quiz 3

1. A kite has a diagonal of 12 and a diagonal of 11.
   What is its area?
2. A rhombus has a diagonal of 2 and a diagonal of
   8. What is its area?
3. A square has a side 20 inches. What is its
   area?
4. A rectangle has a base of 16, and a height of 5.
   What is its area?
5. A parallelogram has a base of 8 and a height of
   8. What is its area?
6. A triangle has a base of 18 and a height of 10.
   What is its area?
7. A triangle has a base of 11 and a height of 8.
   What is its area?
8. A trapezoid has a top base of 8, a bottom base of
   9, and a height of 6. what is its area?
9. A trapezoid has a top base of 4, a bottom base of
   20, and a height of 23. What is its area?
10. A parallelogram has a base of 20 and a height of
    23. What is its area?

54
Unit 7 Quiz 5

• Match the equation with the definition

• Equation

1. Area of a Triangle
2. Area of a Parallelogram or Rectangle
3. Area of a Rhombus or Kite
4. Area of a Trapezoid
5. Area of a Square
6. Length of a side on a 45-45 triangle
7. Length of the hypotenuse on a 30-60-90 triangle
8. Length of the hypotenuse on a 45-45 triangle
9. Length of the short side of a 30-60-90 triangle
10. Length of the long side of a 30-60-90 triangle

1. S (Hv2)/2
2. A S2
3. A (B x H)/2
4. L Sv3
5. A H x (B1 B2)/2
6. S H/2 or S Lv3/3
7. A B x H
8. H 2S
9. A (D1 x D2)/2
10. H Sv2

55
Unit 7 Quiz 4 ?10 Points

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

56
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
57
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

58
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
59
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
60
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
61
Assignment

• Page 672/673 16-24

62
Unit 7 Quiz 6

• Write the equation to calculate p

63
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
64
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

65
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
66
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
67
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

68
Assignment

• Page 638 9-16,19-29
• Worksheet 8-6

69
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?

70
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
71
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
72
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

73
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
74
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
75
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

76
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
77
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

78
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.

79
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
80
Assignment

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

81
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?

82
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

83
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
84
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?

85
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