Geometry 1 February 2013

Warm up: Warm up: Check your homework. Check your homework. For EACH PROBLEM: √ if correct. X if incorrect.For EACH PROBLEM: √ if correct. X if incorrect.Work with your group mates to find and correct any errors. Please use a different color.

Use the HW rubric on the purple sheet and grade yourself. I will revise if necessary!

objectiveStudents will review finding area of circles and parts

of circles and show understanding on a quiz.

Homework due todayHomework due todaypg. 337+: 1 – 12, 21, 22pg. 337+: 1 – 12, 21, 22Finish 8.5/8.6 HandoutFinish 8.5/8.6 Handout

complete problem statement complete problem statement for Kribz projectfor Kribz project

Homework Due TuesdayHomework Due Tuesdaypg. 471: 1 – 8 (incl sketch), 11-13, 16 – 26, 38pg. 471: 1 – 8 (incl sketch), 11-13, 16 – 26, 38

Extra Credit HW- due Feb 1Extra Credit HW- due Feb 1Investigation pg. 449Investigation pg. 449

Briefly summarize what you need to do for each step and clearly write your answers for each

step

Materials: paper, compass, scissors, tape

Tape your pieces to your paper Label the height and length in terms of the

circumference and radius of the circle.What is the formula for your “parallelogram”?

Project DSH Kribz See handout

PROJECT DUE: FEBRUARY 12

summaryAparallelogram = bh

Atriangle = ½ bh

Atrapezoid =½ (b1+b2)h

Akite = ½ (d1)(d2)

Aregular polygon= ½ san = ½ aP

π = C/d C = πd = 2πr Acircle = πr2

2

b hA

Area formulas:

A b h 1 2

2

h b bA

1 2

2

d dA

Regular Polygons

2 2

asn aPA

Circles

A = πr2

C = 2𝝅r = 𝝅d𝝅 = C/d (definition!)

Term Definition Example

Circle sector area

conjecture

The area of a sector of a circle is given by the formula,

A is the area and r is the radius of the circle, and ‘a’ is the degree of the inscribed

angle

Area of a segment of a

circle

see page 453

Area of an annulus of a

circle

see page 453

2 2 2a b c

2

360

aA r

Quiz

Please clear your desks.Work silently.REQUIRED format:a) write formulab) substitutec) do the mathd) units

finished? you may silently work on homework

Honors Geometry 6 February 2012Warm up: 1) Find the height of the trapezoid if A ≈ 256 yd2

a) 1.78 ft b) 8 ft c) 16 ft Show your work to justify your answer.

2) SOLVE for x: x (x – 5) = 40 – 2x

Place your project preliminary proposal on your Place your project preliminary proposal on your desk for a homework check.desk for a homework check.

15 ft

17 ft

ObjectiveStudents will review and apply the Pythagorean

Theorem to solve problems.

Students will take notes, watch a video and use think-pair-share as they solve problems.

Project See handout

PROJECT DUE: FEBRUARY 6

Large is 16 inch- $7.99Medium is 14 inch- $5.99Justify your answer with Math!

Which is the better deal?

Which is the better deal?

Challenge QuestionImagine a steel belt fitting tightly around Earth’s

equator. Now imagine cutting the belt and splicing in a piece to make the belt 40 feet longer. Make the longer belt stand out evenly from the equator. (HINT- Cearth≈ 24901 miles)

What’s the largest object that will fit under the belt: an atom? an ant? a large dog?

an elephant?Explain your answer in complete sentences. You

may make a sketch to help you think about it.

a

b

c

222 cba

This is a right triangle:

We call it a right triangle because it contains a right angle.

The measure of a right angle is 90o

90o

The little square in the angle tells you it is a

90o

right angle.

About 2,500 years ago, a Greek mathematician named Pythagorus discovered a special relationship between the sides of right triangles.

Pythagorus realized that if you have a right triangle,

3

4

5

and you square the lengths of the two sides that make up the right angle,

24233

4

5

and add them together,

3

4

5

2423 22 43

22 43

you get the same number you would get by squaring the other side.

222 543 3

4

5

Is that correct?

222 543 ?

25169 ?

It is. And it is true for any right triangle.

8

6

10222 1086

1006436

video of investigation pg. 478

http://www.youtube.com/watch?NR=1&feature=endscreen&v=uaj0XcLtN5c

Baseball A baseball scout uses many

different tests to determine whether or not to draft a particular player. One test for catchers is to see how quickly they can throw a ball from home plate to second base. The scout must know the distance between the two bases in case a player cannot be tested on a baseball diamond. This distance can be found by separating the baseball diamond into two right triangles.

Right Triangles• Right Triangle – A

triangle with one right angle.

• Hypotenuse – Side opposite the right angle and longest side of a right triangle.

• Leg – Either of the two sides that form the right angle.

Leg

Leg

Hypotenuse

Pythagorean Theorem• In a right triangle, if a and b

are the measures of the legs and c is the measure of the hypotenuse, then

a2 + b2 = c2.• This theorem is used to find

the length of any side of a right triangle when the lengths of the other two sides are known.

b

a

c

Finding the Hypotenuse

• Example 1: Find the length of the hypotenuse of a right triangle if a = 3 and

b = 4. 4

3c

a2 + b2 = c2

5

5

25

25

169

43

2

2

222

c

c

c

c

c

c

Finding the Length of a Leg

• Example 2: Find the length of the leg of the following right triangle.

9

12

a

a2 + b2 = c2

14481

14481

129

2

2

222

a

a

a

81 81__________________

94.7

63

632

a

a

a

Examples of the Pythagorean Theorem

• Example 3: Find the length of the hypotenuse c when a = 11 and b = 4. Solution

• Example 4: Find the length of the leg of the following right triangle.

Solution

11

4

c

5

13a

Solution of Example 3

• Find the length of the hypotenuse c when

a = 11 and b = 4.

a2 + b2 = c2

11

4

c

70.11

137

137

16121

411

2

2

222

c

c

c

c

c

Solution of Example 4

16925

1352

222

222

a

a

cba • Example 4: Find the length of the leg of the following right triangle.

13a

5

2525_______________

12

144

1442

a

a

a

Converse of the Pythagorean Theorem

• If a2 + b2 = c2, then the triangle with sides a, b, and c is a right triangle.

• If a, b, and c satisfy the equation a2 + b2 = c2, then a, b, and c are known as

Pythagorean triples.

Example of the Converse

Example 5: Determine whether a triangle with lengths 7, 11, and 12 form a right triangle.

**The hypotenuse is the longest length.

14412149

12117?

2?

22

144170

This is not a right triangle.

Example of the Converse

Example 6: Determine whether a triangle with lengths 12, 16, and 20 form a right triangle.

400256144

201612?

2?

22

400400 This is a right triangle. A set of integers such

as 12, 16, and 20 is a Pythagorean triple.

Converse ExamplesExample 7: Determine

whether 4, 5, 6 is a Pythagorean triple.

Example 8: Determine whether 15, 8, and 17 is a Pythagorean triple.

362516

654?

2?

22

36414, 5, and 6 is not a Pythagorean triple.

28964225

17815?

2?

22

289289 15, 8, and 17 is a Pythagorean triple.

Baseball Problem• On a baseball diamond, the hypotenuse is

the length from home plate to second plate. The distance from one base to the next is 90 feet. The Pythagorean theorem can be used to find the distance between home plate to second base.

Solution to Baseball Problem

2

222

222

81008100

9090

c

c

cba

• For the baseball diamond, a = 90 and

b = 90.

90

90

2200,16 cc200,16

127cThe distance from home plate to second base is

approximately 127 feet.

c

practiceClassworkdo pg. 481: 1 – 10

be ready to share your work

debrief…can you find the errors?

http://www.youtube.com/watch?NR=1&feature=fvwp&v=fO9EU0w3CrY

IIlustration of proof on pg. 479:http://www.youtube.com/watch?v=pVo6szYE13Y&feature=endscreen&NR=1

Honors Geometry 28 Jan 2012Clean out your group folder.

WARM UP- THINK- 3 minutes silently PAIR- chat with a partner

1. ABCD is a parallelogram. What is the measure of angle D?

a) 22.5⁰ b) 45c) 67.5⁰ d) 112.5⁰

2. Find x and check your answer:

x (x – 2) = 3x + 6 What do the values of x represent on the graph?

(5x)⁰

(3x)⁰

A B

CD

Honors Geometry 31 Jan 2012WARM UP- THINK- 3 minutes silently PAIR- chat with a partner

1. An equilateral triangle is pictured. If the height is doubled, which of the following statements is true?

a) the measures of the base angles increase slightlyb) the measures of the base angles do not changec) the measures of the base angles are doubled

2. Find n.

h

8 1 2 9 3n n

x-box method of basic factoringfind two numbers that multiply to give you the top number and also add to give you the bottom

given ax2 + bx + cac

bn m

find n and m so that nm = ac AND n + m = b

then ax2 + bx + c = (x + n)(x + m)

ac—air conditioninggoes in the “attic”

b goes in the“basement”

using factoring to solve equationsFind x if x2 + 5x + 6 = 0

a) find the factors of the quadraticb) set EACH factor equal to ZERO and solvec) check

(x + 2)(x + 3) = 0

so x + 2 = 0 OR x + 3 = 0either would make the equation true

x = -2 OR x = - 3

(-2)2 + 5(-2) + 6 = 0 4 + -10 + 6 = 0 0 = 0√

(-3)2 + 5(-3) + 6 = 0 9 + -15 + 6 = 0 0 = 0√

Geometric Probability

outcome- a possible result

event- a set of desired outcomes

probability- the chance that something will happen, expressed as a decimal, fraction or %

Probability = -----------------------------------

P(event ) means “probability of an event”

# of desired outcomes

total # of outcomes possible

0 to 1, 0 to 100% If the outcomes are equally likely, probability (event) = # of outcomes interested in total # of possible outcomes

1.Why is the smallest probability = 0?2.Why is the largest probability = 1 or 100% ?3.What does a probability of 2.3 imply?4.Does it matter if probabilities are written as fractions, decimals or percents?

Rug games

Let’s pretend I have a rug at my house, and there is a trap door in the ceiling directly over the rug. The trap door is the same shape and size as the rug. From time to time, the trap door opens and a dart drops directly down onto the rug. The process is quite random, which means that every point of the rug has as good a chance of getting hit as any other.

Now, of course, my guests never sit directly on the rug (it is dangerous!), but they like to sit nearby and guess which part of the rug the next random dart will hit. To keep things interesting, I have a variety of rugs of the same size that I can put out on different occasions.

Look at the first rug. Which color would you predict the dart is most likely to hit?

What is P(gray)? P(white)?

Rug Games

1) Which color is most likely to be hit by a random falling dart?2) Calculate the probability for each color for each rug. Remember, to be equally likely, rugs must be cut into equal size pieces.3) What if white areas are worth 2 points, grey areas worth 3 points and black areas worth 4 points? How many points for each color would you expect to win if you played a lot of games?

Debrief

what is probability?what must be true about the pieces to be able

to calculate probability?how do you calculate probability?

Prove parallelogram area conjecture Prove parallelogram area conjecture using 2-column or flowchart proofusing 2-column or flowchart proof

Given: ABCD is a parallelogram and h is an altitude.

Using Area Formulas

Example 7 Calculate the area of the triangle below:

3

4

A

B C

-Draw an obtuse triangle.-Make a copy of it.-Rearrange both triangles to make a shape for whichyou already know the area.

Geometry 16/17 Jan 2012WARM UP- THINK- 2 minutes silently PAIR- chat with a partner

1. Solve for x: x3 – 4x + 2 = x4 – 10x + 6 a) -1 b) 0 c) 2 d) 5 Explain how you know your answer is correct.

2. Substitute and evaluate if x = -2 (show all steps): 36 – 3(2 x )

| | 1x

area = ½ ( 3 )( 6 ) = 9 square units

area = ½ ( 4 )( 7 ) = 14 square units

area = ½ ( 5 )( 9 ) = 22 ½ square units

area = ½ ( h )( b )

Do Now:1.1. Write the Area formula inside the appropriate figure:Write the Area formula inside the appropriate figure:

4. A garden 4 ft by 8 4. A garden 4 ft by 8 is surrounded by a is surrounded by a

sidewalk 3 feet wide– sidewalk 3 feet wide– Determine the area of Determine the area of

the sidewalkthe sidewalk

BDAB CB

2. A rectangle yard is 20 meters by 44 meters. 2. A rectangle yard is 20 meters by 44 meters. If a rectangular swimming pool 9 meters by 11 If a rectangular swimming pool 9 meters by 11 meters is put in the yard, how much yard area meters is put in the yard, how much yard area

is left?is left?3. The area is 64, find h3. The area is 64, find h

Ac132o3

7 18

x

h

16

Using Properties of Kites

A quadrilateral is a kite if and only if it has two distinct pair of consecutive sides congruent.

• The vertices shared by the congruent sides are ends. • The line containing the ends of a kite is a symmetry line for a kite. • The symmetry line for a kite bisects the angles at the ends of the kite. • The symmetry diagonal of a kite is a perpendicular bisector of the other diagonal.

Using Properties of Kites

A

B C

D

Theorem 6.19

If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent.

mB = mC

Using Properties of Kites

x°

125°

(x + 30)°

A

B C

D

Example 8

ABCD is a kite. Find the mA, mC, mD