Pythagorean Theorem

Vocabulary

1. Leg – One of 2 sides of a right triangle2. Hypotenuse – the longest side of a right triangle3. Triangle- a 3-sided polygon4. Polygon- a 2-D figure, has at least 3 sides5. Equilateral- all sides equal

Vocabulary

6. Isosceles- 2 sides are equal7. Scalene- NO sides are equal8. Pythagorean Theorem- a formula to find a side of a right triangle ( a^2 + b^2 = c^2 )9. Right Angle- a 90 degree angle10. Right Triangle- a triangle with 2 legs, a hypotenuse, and a right angle

Vocabulary11. Degrees- a measurement of an angle represented by a small circle12. Radians- a measurement of an angle represented by a small hash mark (‘)13. Side, Side, Side (SSS)- A method to prove that 2 or more triangles are congruent by having 3 congruent sides14. Angle, Side, Angle (ASA)- A method to prove that 2 or more triangles are congruent by having an angle, a side, and an adjoining angle congruent15. Angle, Angle, Angle (AAA)- A method to prove that 2 or more triangles are congruent by having 3 congruent angles

Vocabulary

16. Congruent- equal17. Similar- close, but not 100% congruent (usually a scale drawing)18. Parallel- are two lines that never cross

19. Perpendicular- are two lines that cross at a right angle20. Straight Angle- a 180 degree line

Vocabulary

21. Complementary- two angles that add to 90 degrees22. Supplementary – two angles that add to 180 degrees23. Perimeter – How much distance is around an object (Fence) (2-D)24. Area – How much it takes to cover an object (Carpet) (2-D)25. Volume – How much it takes to fill an object (3-D)

FormulasPythagorean Theorem c2 = a2 + b2

Perimeter Polygons- P = add all sides

Area Triangle - A = (1/2)bhSquare/Rectangle - A = bhTrapezoid - A = (1/2) (p + q)h

VolumePrisms - V = (BA)hCones - V = (1/3)(BA)hCylinders V = (BA)hSphere V = (4/3)(BA)hHemisphere V = (2/3)(BA)h

Shape Chart - Polygons

Shape Picture Sides Perimeter Area

Triangle 3 P = a + b + c A = (1/2)bh

Square 4 P = 4s A = bh

Rectangle 4 P = 2b + 2h A = bh

Pentagon 5 Add all sides ----------------

Hexagon 6 Add all sides ----------------

Heptagon 7 Add all sides ----------------

Octagon 8 Add all sides ----------------

Nonagon 9 Add all sides ----------------

Decagon 10 Add all sides ----------------

Dodecagon 12 Add all sides ----------------

OOOOOOOOO Circles OOOOOOOOOO

No Perimeter!No Sides!

Circumference -------------- C = 2πr C = dπArea ------------------------- A = πr2

Finding Distance

• Traditional Formula: d = √(X2 – X1) + (Y2 – Y1)

• THERE MUST BE AN EASIER WAY!

Distance With Pythagorean Theorem

1. Draw a triangle2. Determine leg lengths3. Solve for hypotenuse length4. Draw a conclusion

Finding Distance on a Grid- Pg. 6 -10- Pg. 22- Pg. 35- Pg. 39

What Makes a Right Triangle?

Pg. 37

Wheel of Theodorus Investigation

The diagram is named for its creator, Theodorus of Cyrene (sy ree nee), a former Greek colony. Theodorus was a Pythagorean.

The Wheel of Theodorus begins with a triangle with legs 1 unit long and winds around counterclockwise. Each triangle is drawn using the hypotenuse of the previous triangle as one leg and a segment of length 1 unit long as the other leg. To make the Wheel of Theodorus, you need only know how to draw right angles and segments 1 unit long.

Sections to Have in Report

1. Objectives (2 minimum)2. Hypothesis (3 minimum)3. Materials4. Procedure5. Data (Chart, Graph, Equation)6. Observations (3 minimum)7. Calculations8. Conclusion/Errors (4 sentence minimum)9. Extension Problems

Objectives

-Refer to the blue sheet-Use your action words!-Try to pick from the 3 columns farthest to the right

Hypothesis- What are we trying to solve?- What do you think?- What will happen?

Materials

-What will you use to create this design?- Not all designs are created the same!

Procedure

- What are the steps to do this project?

- You can update it as you go if it does not make sense.

DataLeg a Leg b Hypotenuese

Triangle 1

Triangle 2

Triangle 3

Triangle 4

Triangle 5

Triangle 6

Triangle 7

Triangle 8

Triangle 9

Triangle 10

Triangle 11

Observations

-What do you notice?-What elements were hard?-What elements were easy?

Calculations

1. Make a table of all the triangles and the hypotenuse lengths.

2. Graph the data (computer or by hand)3. Develop an equation if possible4. Describe the data (Increasing/Decreasing?

Growth/Decay?, Constant?, etc…)

Conclusions

1. Were your hypothesis right or wrong? Why?2. Did you have any errors? How did they effect

your design?3. What unit of measure did you use for your

triangles?4. Why did you choose to decorate your project the

way you did?

Extension Questions

• For each hypotenuse length that is not a whole number: Give the two consecutive whole numbers the length is between. For example √2 is between 1 and 2.

• Odakota uses his calculator to find √3. He gets 1.732050808. Geeta says this must be wrong because when she multiplies 1.732050808 by 1.732050808, she gets 3.000000001. Why do these students disagree?

Shadow Problems

•Pg. 54 and 55•Problems 8-10

3-D FiguresShape Faces Edges Vertices Volume

Triangular Prism

5 9 6 V = (BA)h

Square Prism

6 12 8 V = (BA)h

Rectangular Prism

6 12 8 V = (BA)h

Trapezoidal Prism

7 15 10 V = (BA)h

Cylinder 3 2 0 V = (BA)h

Cone 2 1 1 V = (1/3)(BA)h

Sphere 1 0 0 V = (4/3)(BA)h

Hemisphere 2 1 0 V = (2/3)(BA)h

Diagonals of Rectangular Prisms

•Pg. 60 •Problem 47 and 48