Geometry 1 February 2013 Warm up: Check your homework. 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!

objective Students will review finding area of circles and parts of circles and show understanding on a quiz. Homework due today pg. 337+: 1 – 12, 21, 22 Finish 8.5/8.6 Handout complete problem statement for Kribz project Homework Due Tuesday pg. 471: 1 – 8 (incl sketch), 11-13, 16 – 26, 38

Extra Credit HW- due Feb 1 Investigation 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

summary A parallelogram = bh A triangle = ½ bh A trapezoid =½ (b 1 +b 2 )h A kite = ½ (d 1 )(d 2 ) A regular polygon = ½ san = ½ aP π = C/d C = πd = 2πr A circle = πr 2

Area formulas: Regular Polygons Circles A = πr 2 C = 2r = d = 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

Quiz Please clear your desks. Work silently. REQUIRED format: a) write formula b) substitute c) do the math d) units finished? you may silently work on homework

Honors Geometry 6 February 2012 Warm up: 1) Find the height of the trapezoid if A ≈ 256 yd 2 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 desk for a homework check. 15 ft 17 ft

Objective Students 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- $7.99 Medium is 14 inch- $5.99 Justify your answer with Math! Which is the better deal?

Challenge Question Imagine 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- C earth ≈ 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

This is a right triangle:

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

The measure of a right angle is 90 o 90 o

The little square in the angle tells you it is a 90 o 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, 3 4 5

and add them together, 3 4 5

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

Is that correct? ? ?

It is. And it is true for any right triangle

video of investigation pg =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 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 a 2 + b 2 = c 2. 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 = c a 2 + b 2 = c 2

Finding the Length of a Leg Example 2: Find the length of the leg of the following right triangle a a 2 + b 2 = c 2 __________________

Examples of the Pythagorean Theorem Example 3: Find the length of the hypotenuse c when a = 11 and b = 4. Solution Solution Example 4: Find the length of the leg of the following right triangle. Solution 11 4 c 5 13 a

Solution of Example 3 Find the length of the hypotenuse c when a = 11 and b = 4. a 2 + b 2 = c c

Solution of Example 4 Example 4: Find the length of the leg of the following right triangle. 13 a 5 _______________

Converse of the Pythagorean Theorem If a 2 + b 2 = c 2, then the triangle with sides a, b, and c is a right triangle. If a, b, and c satisfy the equation a 2 + b 2 = c 2, 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. 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. This is a right triangle. A set of integers such as 12, 16, and 20 is a Pythagorean triple.Pythagorean triple.

Converse Examples Example 7: Determine whether 4, 5, 6 is a Pythagorean triple. Example 8: Determine whether 15, 8, and 17 is a Pythagorean triple. 4, 5, and 6 is not a Pythagorean triple. 15, 8, and 17 is a Pythagorean triple.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 For the baseball diamond, a = 90 and b = The distance from home plate to second base is approximately 127 feet. c

practice Classwork do pg. 481: 1 – 10 be ready to share your work

debrief…can you find the errors? IIlustration of proof on pg. 479:

Honors Geometry 28 Jan 2012 Clean 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) 45 c) 67.5⁰d) 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 C D

Honors Geometry 31 Jan 2012 WARM 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 slightly b) the measures of the base angles do not change c) the measures of the base angles are doubled 2. Find n. h

x-box method of basic factoring find two numbers that multiply to give you the top number and also add to give you the bottom given ax 2 + bx + c ac b nm find n and m so that nm = ac AND n + m = b then ax 2 + bx + c = (x + n)(x + m) ac— air conditioning goes in the “attic” b goes in the “basement”

using factoring to solve equations Find x if x 2 + 5x + 6 = 0 a) find the factors of the quadratic b) set EACH factor equal to ZERO and solve c) check (x + 2)(x + 3) = 0 so x + 2 = 0 OR x + 3 = 0 either would make the equation true x = -2 OR x = - 3 (-2) 2 + 5(-2) + 6 = = 0 0 = 0√ (-3) 2 + 5(-3) + 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 using 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: -Draw an obtuse triangle. -Make a copy of it. -Rearrange both triangles to make a shape for which you already know the area.

Geometry 16/17 Jan 2012 WARM UP- THINK- 2 minutes silently PAIR- chat with a partner 1. Solve for x: x 3 – 4x + 2 = x 4 – 10x + 6 a) -1b) 0c) 2d) 5 Explain how you know your answer is correct. 2. Substitute and evaluate if x = -2 (show all steps):

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.Write the Area formula inside the appropriate figure: 4. A garden 4 ft by 8 is surrounded by a sidewalk 3 feet wide– Determine the area of the sidewalk 2. A rectangle yard is 20 meters by 44 meters. If a rectangular swimming pool 9 meters by 11 meters is put in the yard, how much yard area is left? 3. The area is 64, find h

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 Theorem 6.19 If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent. m  B = m  C

Using Properties of Kites Example 8 ABCD is a kite. Find the m  A, m  C, m  D