Warm-Up (7 min) Full sentences
• Start a new page, date on top.
• Homework can be found on the __________ and
the ____________________ .
• Update planner now.
• The following tools are required every day:___
_________, ________, ________, _________.
• On a test, you may use your own __________.
• At the end of class, you are graded on ______
_______________, ___________, ___________
• If you have been absent, it is your responsibility to
___________________________________.
• Start a new page, date on top.
• Homework can be found on the back wall
and the class webpage . Update planner now.
• The following tools are required every day: pencils,
notebook, grading pen, compass, protractor.
• On a test, you may use your own calculator.
• At the end of class, you are graded on leaving floor
clean, book & whiteboard in desk, chair pushed in.
• If you have been absent, it is your responsibility to
submit late HW after school with note, and take
notes from website or friend.
Ch. 9
The Pythagorean Theorem
Special Right Triangles
Distance in Coordinate Geometry
Ch. 12
Trigonometric Ratios
Right Triangles
Long side, opposite right angle
The Pythagorean Theorem
c is always the hypotenuse.
It does NOT matter which leg is a or b.
Find missing side x.
Find missing side x.
Find missing side x.
Find missing side x.
Find missing side x.
Find missing side y.
Find missing side y.
Find missing side y.
Find missing side y.
Find missing side y.
Find missing side x on your own.
25
x
20
Complete warm-up in notebook (7 min)
• Planner up-to-date? HW due & out? Date in notes?
• You must have a red or green pen to correct your
own _____ and other people’s _________ when
we go over every answer.
• Tutoring is available in this room ___________ and
_____________ most weeks.
• Quizzes may be retaken after ______within _____.
• To improve a unit test grade, complete the
necessary exercises on ____________________ .
• Extra credit is available by _______________ and
_________________________________________
• You must have a red or green pen to correct
your own HW and other people’s quizzes when
we go over every answer.
• Tutoring is available in this room Tuesday and
Thursday most weeks.
• Quizzes may be retaken after school within one
week of the original.
• To improve a unit test, complete the necessary
exercises on Khan Academy.
• Extra credit is available by winning games and
explaining challenging solutions to the class.
Team Activity (10 min)
1. Cut out six squares from graph paper:
3, 4, 5, another 5, 12, & 13 units per side.
2. Arrange sets of three squares, corner-touchingcorner, so that they form right triangles.
3. With your team, discuss how these
arrangements demonstrate the Pythagorean
Theorem.
4. Draw both arrangements in your notes, label
with dimensions, explain how they demonstrate
the P.T. in words and equations.
Can you make a right triangle from
squares with 9, 11 & 14 cm?
Pythagorean Theorem
If a shape is a right triangle,
then a2 + b2 = c2.
Converse of the Pythagorean
Theorem
If a2 + b2 = c2, then a shape is a
right triangle.
Converse of Pythagorean Theorem
112 + 92 ? 142
121 + 81 ? 196
202 ≠ 196
Not a right
triangle!
Sides do not work
with the P.T.
Pythagorean Theorem
If a shape is a right triangle, then a2 + b2 = c2.
Converse of the Pythagorean Theorem
If a2 + b2 = c2, then a shape is a right triangle.
Inverse of the Pythagorean Theorem
If a shape is not a right triangle, then a2 + b2 ≠ c2.
Contrapositive of the Pythagorean Theorem
If a2 + b2 ≠ c2, then a shape is not a right triangle.
Any HW to put out? Planner up-to-date?
Copy this…
• A right triangle with sides of 3 and 4 units
long, always has a hypotenuse of exactly
5 units.
10
x2
8
6
• This 3-4-5 ratio is the same even if
dilated.
Special Right Triangles
• All triangles with two congruent
corresponding pairs of angles are similar.
• That’s the AA Triangle Similarity Shortcut.
Special Right Triangles
• Because the sides all grow by the same scale
factor, the ratio of sides within the triangle
remains the same. Some special ratios are:
1.41 ≈
1.73 ≈
• Are these right triangles?
• Are these special right triangles?
• If so, which ones? Justify your reasoning.
9 cm
26 cm
13 cm
15 cm
10 cm
24 cm
Calculate the length of this segment
(4, 5)
(2, 2)
1. Calculate lengths of ‘legs’
(2, 2)
(4, 5)
3 units
2 units
X=4–2=2
Y=5–2=3
Δ = change
Δ = final - initial
Δx = x2 – x1
Δy = y2 – y1
2. Calculate segment length with P.T.
X=4–2=2
(4, 5) Y = 5 – 2 = 3
3 units a2 + b2 = c2
(2, 2)
2 units
22 + 32 = c2
√(22 + 32) = √c2
√(4 + 9) = √c2
√13 = c
3.6 units ≈ c
Calculate the length of this segment
Calculate the length of this segment
Calculate the slope of this segment
Calculate the slope of this segment
Calculate the slope of this segment
Calculate the midpoint of this segment
Calculate the midpoint of this segment
Calculate the midpoint of this segment
This is the bottom half of a segment.
Calculate the upper left endpoint,
and its overall length.
This is the bottom half of a segment.
Calculate the upper left endpoint,
and its overall length.
In Your Notebook (10 min)
1. Graph (1, 1) (2, 1) (1, 2)
2. Use rays centered on the origin and going
through each vertex, dilate this triangle with
a scale factor of 4.
3. Write a coordinate rule for this dilation.
4. Using your protractor, measure and label all
interior angles of the pre-image and image.
5. Using a ruler, measure and label all the side
lengths in both triangles.
Story Problems Handout Right Triangle Geometry
• Make sure you get each question
stamped. Due before end of class.
• For each question, each part, draw
detailed pictures. Label distances with
units.
• #1 picture & calculations at bottom.
• #2-4 on back.
Warm-Up (5 min)
• What are the missing side lengths?
6m
?m
?m
?m
1m
?m
Warm-Up Answer
6m
2m
√3 m
3√3 m
1m
3m
Ratios of sides stay the same
when dilating or scaling up/down
6m
2m
√3 m
3√3 m
1m
3m
Trigonometry Ratios
• Sine of angle = __Opposite length___
Hypotenuse length
• Cosine of angle = __Adjacent length___
Hypotenuse length
• Tangent of angle = _Opposite length___
Adjacent length
Trigonometry Ratios Mneumonic
• SOH = Sin(e) angle = __Opposite length___
Hypotenuse length
• CAH = Cos(ine) angle = __Adjacent length___
Hypotenuse length
• TOA = Tan(gent) angle = _Opposite length___
Adjacent length
Trigonometry Ratios
• Sin angle = __Opposite length___
Hypotenuse length
hypotenuse
opposite
• Cos angle = __Adjacent length_
Hypotenuse length
• Tan angle = _Opposite length_
Adjacent length
angle
adjacent
Intro to Trigonometric Ratios
https://www.khanacademy.org/math/geometry
/right-triangles-topic/intro-to-the-trig-ratiosgeo/v/basic-trigonometry
Calculate the sine of 30o for both.
Calculate the cosine of 30o for both.
Calculate the tangent of 30o for both.
6m
2m
√3 m
3√3 m
1m
3m
Intro to Trigonometric Ratios Part II
https://www.khanacademy.org/math/geometry
/right-triangles-topic/intro-to-the-trig-ratiosgeo/v/basic-trigonometry-ii
Get out your cell phone or scientific calculator.
• On some calculators, enter degrees
of angle, then hit sin, cos or tan.
• On other calculators, hit sin, cos or
tan first, then degrees of angle. Then
enter.
• Write in notebook: sin 30o = ____,
cos 30o = ____ and tan 30o = _____.
Fill in answers. Check with partner.
Solving for a Side with Trig Ratios
https://www.khanacademy.org/math/geometry
/right-triangles-topic/trig-solve-for-a-sidegeo/v/example-trig-to-solve-the-sides-andangles-of-a-right-triangle
Find the missing leg length.
Which Trigonometry Ratio to Use?
53o
10 m
Find hypotenuse length.
Which Trigonometry Ratio to Use?
53o
10 m
5 min Warm-Up
Draw and label diagram.
Find missing side lengths.
Which Trigonometry Ratios to Use?
36o
10 m
Have you updated your planner today?
Inverse Trig Ratios
Draw and label diagram.
Find missing angle.
2m
3m
?o
Trigonometry Practice
• Two practice worksheets each.
• Work with partners who have same sheet.
• When you complete the first worksheet, get
checked off for credit.
• Receive second, complete, get checked off.
• HW due tomorrow as well.
Have you updated your planner today?
Trigonometry Practice
•
•
•
•
HW out for credit in first minute.
Two practice worksheets each.
Work with partners who have same sheet.
When you complete the first worksheet, get
checked off for credit.
• Receive second, complete, get checked off.
Have you updated your planner today?
Right Triangles Review Warm-Up (5 min)
1. In a right triangle, if you know the lengths of
two sides, you can calculate the third using
the __________________.
2. In any right triangle, the _____ of an angle is
equal to the ratio of the opposite
side/hypotenuse.
3. A right triangle has a ‘leg’ of 6 and a
hypotenuse of 10. Calculate the missing leg.
4. Calculate the three interior angles in the
triangle in Q3?
Whack-a-Mole Review Game
•
•
•
•
•
•
•
•
•
Teams of 3-4 at adjacent desks.
All your names on answer sheet.
Number answer sheet 1-25.
Each team starts with a few question cards.
Answer these in correct place on sheet.
Hold up cards to signal ‘ready-to-trade.’
One person each team makes trade.
When answer all 25 Q’s, turn in sheet, return.
If some wrong, will be returned. Trade for those
cards and revise answers.
• Extra credit based on when you get all correct.
1. HW out for check.
2. Take trigonometry quiz.
3. Switch ‘n’ grade quiz.
4. Go over HW.
5. Any missing quizzes/HWs in
gradebook? Complete after school
today.
6.Junior SAT tomorrow.
7.Launch rocket tomorrow. Max alti?