11-17-2025 Pythag & Triangle Review and Trig

Agenda
Bell Work
Mood Meter
Objectives
Triangle Basis Review
Trig Basics
Practice, Practice, Practice

​Take out paper or device for notes

Open Ended

Choose your question, then share your response.

Names and responses will be shared.

Poll

Which artwork depicts your mental state today?

• 1

• 2

• 3

• 4

• 5

What's next?

I can apply the Pythagorean Theorem in a right triangle to find a missing side.

• I can combine trig ratios and the Pythagorean Theorem to solve for a missing side or angle in a right triangle.

• I can interpret and solve applied (word-problem) situations involving right triangles using trig and Pythagorean reasoning.

• I can verify solutions by checking consistency between trig ratio, Pythagorean Theorem, and context.
  I can explain how similarity in right triangles leads to definitions of sine, cosine, and tangent.

• I can identify opposite, adjacent, and hypotenuse sides in a right triangle for a given acute angle.

• I can compute the sine, cosine, or tangent of an acute angle in a right triangle.

• I can solve for a missing side or angle in a right triangle using trig ratios.

• I can apply right-triangle trigonometry to real-world problem contexts (e.g., heights, inclines).

M.GHS.21: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
M.GHS.19: Define trigonometric ratios and solve problems involving right triangles.

​Take out paper/pencil/pen, markers for notes

Classifying ​Triangles Review

​Breakout Rooms

• Press the PLAY BUTTON to listen to audio of slides

• Advance slides at your own pace

• Share your screen

• Write your notes

• Take your time on practice problems-it's a grade

• Let me know if you have questions

​Check for Understanding

Match

Match the following triangles with their side classification.

Equilateral Triangle

Scalene Triangle

Isosceles Triangle

• Equilateral Triangle

• Scalene Triangle

• Isosceles Triangle

Match

Match the following triangles with their angle classification.

Right Triangle

Acute Triangle

Obtuse Triangle

• Right Triangle

• Acute Triangle

• Obtuse Triangle

Match

Match the following triangles with their side and angle classification.

Equilateral

Scalene Obtuse

Scalene Right

Isosceles Acute

• Equilateral

• Scalene Obtuse

• Scalene Right

• Isosceles Acute

Drag and Drop

Side Classification: ​

Placeholder

Angle Classification: ​

Placeholder

Drag these tiles and drop them in the correct blank above

Isosceles

Acute

Right

Obtuse

Equilateral

Scalene

Drag and Drop

Side Classification: ​

Placeholder

Angle Classification: ​

Placeholder

Drag these tiles and drop them in the correct blank above

Scalene

Right

Isosceles

Acute

Obtuse

Equilateral

Drag and Drop

Side Classification: ​

Placeholder

Angle Classification: ​

Placeholder

Drag these tiles and drop them in the correct blank above

isosceles

right

scalene

obtuse

equilateral

acute

THE PYTHAGOREAN THEOREM AND ITS CONVERSE

1. ​1. Write your equation

2. Square your numbers

3. Balance the equation

   Combine Like Terms

4. Take the Square Root​

5. ​IF necessary simply the root

Solving for the hypotenuse

1. ​1. Write your equation

2. Square your numbers

3. Balance the equation

   Combine Like Terms

4. Take the Square Root​

5. ​IF necessary simply the root

Solving for a missing leg

​Check for Understanding

Dropdown

The hypotenuse of a right triangle measures 17 cm.

One of its legs measures 15 cm.

The other leg measures 9 cm.

This ​

Select answer

a right triangle.

Multiple Choice

_{^{Find the missing side}}

• 48
• 108
• 83.6
• 72

Multiple Choice

Which equation would be used to solve for x?

• x²+31²=67²
• tan(31)= x/67
• 67²+31²=x²
• cos^-1(31/67)

Multiple Choice

What is the length of the hypotenuse in the triangle shown?

• 3

• 4

• 5

• 8

Multiple Choice

Level 4

Solve for x.

• 38.9

• 40.2

• 45.6

• 54

Multiple Choice

Square roots are the opposite of ____________.

• Cube Roots
• Squaring
• Multiplication
• Absolute Value

Multiple Choice

In a right triangle, if a = 8 and c = 17, what is the correct setup to find the value of b?

• $$17^2+8^2=b^2$$

• $$8^2+b^2=17^2$$

• $$17^2+b^2=8^2$$

• $$\frac{8^2}{b^2}=\frac{17^2}{100}$$

Multiple Choice

What is the final step in solving this work for c?
a² + b² = c²
3² + 4² = c²
9 + 16 = c²
25 = c²

• Divide both sides by 2
• Take the square root of both sides

Multiple Choice

An airplane flies 56 miles due north and then 33 miles due east. How many miles is the plane from its starting point?

• 65 miles
• 89 miles
• 45.4 miles
• We don't have enough information.

Multiple Choice

What is the approximate length of the kite and its string from the top of the tree to where it rests on the ground? Use pythagorean theorem to solve. 

• 14 feet

• 22 feet

• 26.08 feet

• 31.91 feet

Multiple Choice

The longest side of a right triangle is DIRECTLY ACROSS from the...

• right angle.
• acute angle.
• obtuse angle.

Multiple Choice

Casey travels from her house directly west to the bank and then directly north from the bank to the mall. She then travels home on the road connecting the mall and her house. What is her total distance traveled?

• 17 miles
• 32 miles
• 37 miles
• 40 miles

ALWAYS be sure the side lengths you are given will actually create a triangle!

Classifying Right Triangles

• If c² = a² + b² --> right triangle

• If c² < a² + b² --> acute triangle

• If c² > a² + b² --> obtuse triangle

​Check for Understanding

Multiple Choice

Determine whether the given side lengths form a triangle. If they do, classify the triangle as acute, right, or obtuse.

31, 35, 39

• acute

• right

• obtuse

• not a triangle

Multiple Choice

Determine whether the given side lengths form a triangle. If they do, classify the triangle as acute, right, or obtuse.

8, 15, 23

• acute

• right

• obtuse

• not a triangle

Multiple Choice

State if the triangle is acute, obtuse or right. Why?

• Obtuse because 6²+7²<10²

• Obtuse because 6²+7²>10²

• Acute because 6²+7²>10²

• Acute because 6²+7²<10²

• Right because 6²+7²=10²

Multiple Choice

Isabella has sticks that are 10cm, 11cm, and 13cm long.  Can she place the sticks together to form a right triangle?  Justify your answer.

• No, because 10+11 does not equal 13
• No, because 100+121 does not equal 169
• Yes, because there is a small side, medium side and long side
• Yes, because a triangle has 3 sides

Multiple Choice

If three sides of a right triangle are 50, 40, and 30, which side is the hypotenuse?

• 50
• 40
• 30

Multiple Choice

Alan made a small four-sided table for his office.  The opposite sides of the table are 27 inches long and 36 inches long.  If the diagonal of the table measures 40 inches, does the table have right angles at the corners?  Why or why not?

• Yes, because it's a rectangle and rectangles have 90 degree angles in the corners
• Yes, because 27 and 36 are less than 40
• No, because 27²+36² does not equal 40²
• No, because 27+36 does not equal 40

Categorize

Options (6)

3,4,5

9,12,15

12,14,15

8,12,13

6,8,11

5,10,13

Determine if each set of side measures would make an acute, right, or obtuse triangle.

Acute

Right

Obtuse

Categorize

Options (6)

9.6, 13.7, 16

4.8, 28.6, 29

12, 18.2, 21.8

1.6, 6.3, 6.5

3.5, 10, 12.5

2.8, 9.6, 12

Determine if each set of side measures would make an acute, right, or obtuse triangle.

Acute

Right

Obtuse

Multiple Choice

What type of triangle is constructed with sides 16, 30, 34?

• Right triangle

• Obtuse triangle

• Acute triangle

• Sides do not form a triangle

Multiple Choice

What type of triangle is constructed with sides 9, 3, 9?

• Right triangle

• Obtuse triangle

• Acute triangle

• Sides do not form a triangle

Multiple Choice

a = 2.1, b = 7.2,
c = 7.5
Is this a right triangle?

• yes 
• no

Poll

I know how to solve 45-45-90 and 30-30-90 triangles.

• Yes I can solve both 45-45-90 and 30-30-90 triangles.

• I can solve on45-45-90 triangles

• I can solve 30-30-90 triangles.

• I cannot solve either 45-45-90 or 30-30-90 triangles.

WRITE THIS DOWN!

Short video on how to solve
45-45-90 Triangles


Draw this triangle on your paper four times - leave at least 4 lines between each pic

Click the link:

https://www.youtube.com/watch?v=dfEPLKYQhz8

​Go to Canvas
Upload a picture of your 45-45-90 notes

Labelling

Label the correct ratios for the following triangle.

Drag labels to their correct position on the image

Multiple Choice

What is the length of y in this picture?

• 45
• 5√2
• 90
• 5

Multiple Choice

What is the length of y in this 45-45-90 triangle?

• 8√2
• 4√2
• 4
• 8

Multiple Choice

What is the length of x in this 45-45-90 triangle?

• 4√2
• 4
• 8
• 4√3

Multiple Choice

Find the value of y.

• 11√2
• 11
• 22
• (11√2)/2

Multiple Choice

Find the lengths of the other two sides of a right triangle if the length of the hypotenuse is 4√2 inches and one of the angles is 45∘.

• 4 inches
• 2 inches
• 8 inches
• 10 inches

Multiple Choice

Sam's square bedroom has a diagonal of 9√2 feet.  What is the length of each side?

• 18
• 3√2
• 9
• 12.7

Multiple Choice

What is the length of y in this picture?

• 45
• 5√2
• 90
• 5

WRITE THIS DOWN!

Short video on how to solve 30-60-90 Triangles


Draw this triangle on your paper eight times. Leave at least 5 lines between each pic


Click the link:

https://youtu.be/90hDli8irfA?si=e6lN_43pYJFXMqD1

​Go to Canvas
Upload a picture of your 45-45-90 notes

​Check your understanding

Match

Match the following for finding sides in a 30-60-90 right triangle.

• Short Leg

• Hypotenuse

• Long Leg

Half of the hypotenuse

Double the short leg

Short leg times square root of 3

• Half of the hypotenuse

• Double the short leg

• Short leg times square root of 3

Labelling

Label the correct ratios for the following triangle.

Drag labels to their correct position on the image

Drag and Drop

Find the value of each variable.

x =​

Placeholder

y = ​

Placeholder

Drag these tiles and drop them in the correct blank above

Fill in the Blanks

Multiple Choice

What is the length of u and v in this 30-60-90 triangle?

• u = 8 v = 4√3
• u = 4√2 v = 8
• u = 16 v = 8
• u = 4√3 v = 8

Multiple Choice

Find a.

• 3

• $$3\sqrt{3}$$

• 6

• $$\sqrt{3}$$

Multiple Choice

Find x.

• 10

• $$10\sqrt{3}$$

• $$10\sqrt{2}$$

• 20

Multiple Choice

Which side is the long leg in this 30-60-90 triangle? (not the hypotenuse)

• 4
• u
• v

Math Response

What is the value of x? Value only, no x =

Type answer here

Correct Answer: 6\sqrt{3}

Deg^°

Rad

Labelling

Label the correct ratios for the following triangle.

Drag labels to their correct position on the image

Poll

What is your confidence level on today's lessons?

• Super Confident

• Confident

• I may have a few questions

• I'm lost!

Home

Open in new tab

Copy link