1. Geometry 15 Jan 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! 

2. Objective
Students will review derivations and apply formulas for area of geometric shapes to solve problems. Students will take notes, work with their group to solve problems, and present work to the class.

3. Homework due Today
1) 8.2, pg : 1 – 15 odds, 19, 20, 22, 29 2) Study and KNOW the area formulas for rectangles, parallelograms, triangles, trapezoid and kites 3) CHOOSE two of the 5 shapes listed above AND explain HOW WE DERIVED THE FORMULAS (you may use a sketch to help!)

4. and solving basic problems
Homework DUE Friday
1) pg. 435: 1, 5, 9
2) pg. 472: 17, 20, 21, 22
3) study for quiz on area formulas for rectangles, parallelograms, triangles, trapezoids, kites
and solving basic problems

5. Area Formulas

6. Rectangle

7. What is the area formula?
Rectangle
What is the area formula?

8. What is the area formula?
Rectangle
What is the area formula? bh
We can develop this formula by drawing
a rectangle on a grid and counting the squares.
Multiplying the number of squares in each row by
the # of rows is simply “fast counting”!

9. What is the area formula?
Rectangle
What is the area formula? bh
What other shape has 4 right angles?

10. What is the area formula?
Rectangle
What is the area formula? bh
Square!
What other shape has 4 right angles?

11. What is the area formula?
Rectangle
What is the area formula? bh
Square!
What other shape has 4 right angles?
Can we use the same
area formula?

12. What is the area formula?
Rectangle
What is the area formula? bh
Square!
What other shape has 4 right angles?
Can we use the same
area formula?
Yes

13. Practice!
17m
Rectangle
10m
Square
14cm

14. Answers Rectangle A = bh A = (10)(17) A =170 m2 Square A = bh
A =196 cm2
14cm

15. So then what happens if we cut a rectangle in half?
What shape is made?

16. So then what happens if we cut a rectangle in half?
Triangle
So then what happens if we cut a rectangle in half?
What shape is made?

17. So then what happens if we cut a rectangle in half?
Triangle
So then what happens if we cut a rectangle in half?
What shape is made?
2 Triangles

18. Triangle So then what happens if we cut a rectangle in half?
What shape is made?
2 Triangles
So then what happens to the formula?

19. Triangle So then what happens if we cut a rectangle in half?
What shape is made?
2 Triangles
So then what happens to the formula?

20. Triangle So then what happens if we cut a rectangle in half?
What shape is made?
2 Triangles bh
So then what happens to the formula?

21. Triangle So then what happens if we cut a rectangle in half?
What shape is made?
2 Triangles bh 2
So then what happens to the formula?

22. Practice!
Triangle
14 ft
5 ft

23. Answers
Triangle
14 ft
A = ½ bh
A = ½ (14)(5)
A = 35 ft2
5 ft

24. Let’s look at a parallelogram.
What happens if we slice off the slanted parts on the ends?

25. Let’s look at a parallelogram.
What happens if we slice off the slanted parts on the ends?

26. Let’s look at a parallelogram.
What happens if we slice off the slanted parts on the ends?

27. Let’s look at a parallelogram.
What happens if we slice off the slanted parts on the ends?

28. Let’s look at a parallelogram.
What happens if we slice off the slanted parts on the ends?

29. Let’s look at a parallelogram.
What happens if we slice off the slanted parts on the ends?

30. Let’s look at a parallelogram.
What happens if we slice off the slanted parts on the ends?

31. Let’s look at a parallelogram.
What happens if we slice off the slanted parts on the ends?

32. Let’s look at a parallelogram.
What happens if we slice off the slanted parts on the ends?

33. Let’s look at a parallelogram.
What happens if we slice off the slanted parts on the ends?

34. Let’s look at a parallelogram.
What happens if we slice off the slanted parts on the ends?
What will the area formula be now that it is a rectangle?

35. Let’s look at a parallelogram.
What happens if we slice off the slanted parts on the ends?
What will the area formula be now that it is a rectangle? bh

36. Parallelogram
Be careful though! The height has to be perpendicular from the base, just like the side of a rectangle! bh

37. Parallelogram
Be careful though! The height has to be perpendicular from the base, just like the side of a rectangle! bh

38. Parallelogram
Be careful though! The height has to be perpendicular from the base, just like the side of a rectangle! bh

39. Rhombus
The rhombus is just a parallelogram with all equal sides! So it also has bh for an area formula. bh

40. Practice!
9 in
Parallelogram
3 in
Rhombus
2.7 cm
4 cm

41. Answers A = bh A = (9)(3) A = 27 in2 Parallelogram Rhombus A = bh
2.7 cm
A = bh
A = (4)(2.7)
A = 10.8 cm2
4 cm

42. Let’s try something new with the parallelogram.

43. Let’s try something new with the parallelogram.
Earlier, you saw that you could use two trapezoids to make a parallelogram.

44. Let’s try something new with the parallelogram.
Earlier, you saw that you could use two trapezoids to make a parallelogram.
Let’s try to figure out the formula since we now know the area formula for a parallelogram.

45. Trapezoid

46. Trapezoid

47. Trapezoid
So we see that we are dividing the parallelogram in half. What will that do to the formula?

48. Trapezoid
So we see that we are dividing the parallelogram in half. What will that do to the formula? bh

49. Trapezoid
So we see that we are dividing the parallelogram in half. What will that do to the formula? bh 2

50. But now there is a problem. What is wrong with the base?
Trapezoid
But now there is a problem.
What is wrong with the base? bh 2

51. Trapezoid
So we need to account for the split base, by calling the top base, base 1, and the bottom base, base 2. By adding them together, we get the original base from the parallelogram. The heights are the same, so no problem there. bh 2

52. Trapezoid
So we need to account for the split base, by calling the top base, base 1, and the bottom base, base 2. By adding them together, we get the original base from the parallelogram. The heights are the same, so no problem there.
base 2
base 1
base 1
base 2
(b1 + b2)h 2

53. Practice!
3 m
Trapezoid
5 m
11 m

54. Answers Trapezoid A = ½ h (b1 + b2) A = ½ (5)(3 + 11) A = 35 m2 3 m

55. Don’t forget ORDER OF OPERATIONS
Using Properties of Trapezoids
When working with a trapezoid, the height may be measured anywhere between the two bases.  Also, beware of "extra" information.  The 35 and 28 are not needed to compute this area.
Area of trapezoid =
Don’t forget ORDER OF OPERATIONS
P E M/D A/S
Find the area of this trapezoid.
A = ½ * 26 * ( )
A = 806 u2

56. Using Properties of Trapezoids
Example 2
Find the area of a trapezoid with bases of 10 in and 14 in, and a height of 5 in.
A = ½ h (b1+ b2)
A = ½ (5)( )
A = ½ (5)(24)
A = 60 in2

57. Using Properties of Trapezoids
Midsegment of a Trapezoid – segment that connects the midpoints of the legs of the trapezoid.

58. Theorem 6.17 Midsegment Theorem for Trapezoids
Using Properties of Trapezoids
Theorem Midsegment Theorem for Trapezoids
The midsegment of a trapezoid is parallel to each base and its length is one-half the sum of the lengths of the bases.
½ (b1+b2) is the length of the midsegment!!

59. Using Properties of Trapezoids
Example 4
Find AB, mA, and mC

60. So there is just one more left!
Let’s go back to the triangle.
A few weeks ago you learned that by reflecting a triangle, you can make a kite.

61. So there is just one more left!
Kite
So there is just one more left!
Let’s go back to the triangle.
A few weeks ago you learned that by reflecting a triangle, you can make a kite.

62. Kite
Now we have to determine the formula. What is the area of a triangle formula again?

63. Kite
Now we have to determine the formula. What is the area of a triangle formula again? bh 2

64. Fill in the blank. A kite is made up of ____ triangles.
Now we have to determine the formula. What is the area of a triangle formula again? bh 2
Fill in the blank. A kite is made up of ____ triangles.

65. Kite
Now we have to determine the formula. What is the area of a triangle formula again? bh 2
Fill in the blank. A kite is made up of ____ triangles.
So it seems we should multiply the formula by 2.

66. Kite bh bh
*2 = 2

67. Kite bh bh
*2 = 2
Now we have a different problem. What is the base and height of a kite? The green line is called the symmetry line (or a diagonal), and the red line is half the other diagonal.

68. Let’s use kite vocabulary instead to create our formula.
Symmetry Line (or a diagonal) *Half the Other Diagonal
A = ½ d1 d2

69. Using Properties of Kites

70. Practice!
Kite
2 ft
10 ft

71. Answers
Kite
A = ½ d1d2
A = (2)(10)
A = 20 ft2
2 ft
10 ft
2 ft

72. Area Kite = one-half product of diagonals
Using Properties of Kites
Area Kite = one-half product of diagonals

73. Find the area of the Kite.
Using Properties of Kites
Example 6
ABCD is a Kite.
Find the area of the Kite. 2 4 4 E 4
A = ½ d1d2
A = ½ (4 + 4)(2 + 4)
A = ½ (8)(6)
A = 24 u2

74. Final Summary Make sure all your formulas are written down!bh bh
(b1 + b2)h 2 2
½ d1* d2

75. practice- do 8.1 handout
Do all problems. Show work that explains your thinking, clearly and concisely (to the point), and easy for SOMEONE ELSE to follow!  Finished? Work on your homework.

76. debrief
how can you relate each shape to a rectangle? how did you find the formula for area of a triangle? trapezoid? kite? what will you do to help yourself remember the formulas?