1. Geometry

2. Pythagorean Theorem Distance Formula Quadrilaterals Midpoint
Geometry
Pythagorean Theorem
Distance Formula
Quadrilaterals
Midpoint
Area and Volume

3. Pythagorean Theorem
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Table of Contents

4. Recall... right triangle is a triangle with a right (90o) angle.
leg
hypotenuse
The sides form that right angle are the legs.
The side opposite the right angle is the hypotenuse.
The hypotenuse is also the longest side.

5. Pythagorean Theorem (R1)
In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.
leg2 + leg2 = hypotenuse2 or
a2 + b2 = c2 a b c

6. Is the missing side a leg or the hypotenuse of the right triangle?
Example
Find the length of the missing side of the right triangle. x 9 12
Is the missing side a leg or the hypotenuse of the right triangle?
hypotenuse

7. x 9 12
= x2
= x2
225 = x2
15 = x
-15 is a extraneous solution, a distance can not equal a negative number.
x = 15

8. Is the missing side a leg or the hypotenuse of the
Example
Find the length of the missing side. x 28 20
Is the missing side a leg or the hypotenuse of the
right triangle?
leg

9. x 28 20
x = 282
x = 784
x2 = 384
x = 19.60

10. The safe distance of the base of the ladder from a wall it leans against should be one-fourth of the length of the ladder.
Thus, the bottom of a 28-foot ladder should be 7 feet from the wall. How far up the wall will a the ladder reach?
28 feet
7 feet ?

11. The ladder will reach 18.3 feet up the wall safely.?
a2 + b2 = c2
72 + b2 = 282
49 + b2 = 384
b2 = 335 b
The ladder will reach 18.3 feet up the wall safely.

12. Try this...
The dimensions of a high school basketball court are 84' long and 50' wide. What is the length of from one corner of the court to the opposite corner? 84 50 x
= x2
9556 = x2
97.75 = x
The court is feet
Answer

13. Pythagorean Theorem Applications
The Pythagorean Theorem can also be used in figures that contain right angles.

14. Find the area of the triangle.
Example
Find the area of the triangle.
13 feet
10 feet
A = bh 1 2
The base of the triangle is given, but we need to find the height of the triangle.

15. By definition, the altitude (or height) of an isosceles triangle is the perpendicular bisector of the base.
13 feet
5 feet h
52 + h2 = 132
25 + h2 = 169
h2 = 144
h = 12
A = (10)(12)
A = (120)
A = 60 feet 1 2

16. Find the perimeter of the rectangle.
Try this...
Find the perimeter of the rectangle.
8 in
10 in
Prect = 2l + 2w
x = 102
x = 6
Prect = 2(6) + 2(8)
P = 28 inches
ANSWER

17. Converse of the Pythagorean Theorem (R2)
If the square of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle. a b c A B C
If c2 = a2 + b2, then
ABC is a right triangle.

18. Remember c is the longest side
Example
Tell whether the triangle is a right triangle.
Remember c is the longest side D E F 7 24 25
c2 = a2 + b2
252 =
625 =
625 = 625
DEF is a right triangle.

19. The missing side is the ________ of the right triangle.1
The missing side is the ________ of the right triangle. A
leg 6 9 x B
hypotenuse

20. Find the length of the missing side.2
Find the length of the missing side. 6 9 x

21. The missing side is the _________ of the right triangle.3
The missing side is the _________ of the right triangle. A
leg x 15 36 B
hypotenuse

22. Find the length of the missing side.4
Find the length of the missing side. x 15 36

23. 5
A NBA court is 50 feet wide and the length from one corner of the court to the opposite corner is feet. How long is the court?
(Round the answer to the nearest whole number) A
94.03 feet B
117.7 feet C
118 feet D
94 feet

24. Find the area of the rectangle.6
Find the area of the rectangle. A
120 feet B
84 feet
8 feet
17 feet C
46 inches D
46 feet

25. Find the area of the triangle.7
Find the area of the triangle.
7 inches
24 inches

26. Distance Formula
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27. Computing the distance between two points in the plane is an application of the Pythagorean Theorem for right triangles.
Computing distances between points in the plane is equivalent to finding the length of the hypotenuse of a right triangle.

28. Relationship between the Pythagorean Theorem & Distance Formula
(x1, y1)
(x2, y1)
(x2, y2)
The distance formula
calculates the distance
using point's coordinates. c
Relationship between the
Pythagorean Theorem & Distance Formula b a
The Pythagorean Theorem states a relationship among the sides of a right triangle.
c2= a2 + b2
The Pythagorean Theorem is true for all right triangles. If we know the lengths of two sides of a right triangle then we know the length of the third side.

29. Distance The Distance Formula (x1, y1) (x2, y2)
The distance between two points, whether on a line or in a coordinate plane, is computed using the distance formula.
The Distance Formula
The distance 'd' between any two points with coordinates
and is given by the formula:
(x1, y1)
(x2, y2)
d =
Note: recall that all coordinates are (x-coordinate, y-coordinate).

30. Example (x1, y1) (x2, y2) Calculate the distance
from Point K to Point I
(x1, y1)
(x2, y2)
d =
Plug the coordinates into the distance formula
Label the points - it does not matter
which one you label point 1 and point 2.
Your answer will be the same.
KI = =
= 4.12

31. Calculate the distance from Point J to Point K
Calculate the distance from H to K

33. Distance Application - Perimeter
The vertices of a triangle are (3, -1), (-2, 4) and (5, 6). Find the perimeter.

34. Distance Application - Circles
A circle is centered center at (3, 4) and has a radius of 3 units. Is (5, 2) on, inside of, or outside of the circle?

35. 8
Calculate the distance from Point G to Point K A
6.32
10.95
7.28
7.62 B C D

36. 9
Calculate the distance from Point I to Point H A
4.24 9
3.46 B C D

37. 10
Calculate the distance from Point G to Point H A 16
4.12
2.24
16.03 B C D

38. Quadrilaterals
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40. Parallelogram Rhombus opposite sides parallel opposites sides parallel
opposites sides congruent all sides congruent

41. Parallelogram Rectangle
opposite sides parallel opposites sides parallel
opposites sides congruent opposite sides congruent
sides are perpendicular (right angles)

42. opposite sides parallel one pair of opposite sides parallel
Square Trapezoid
opposite sides parallel one pair of opposite sides parallel
all sides congruent
sides are perpendicular (right angles)

43. The vertices of a quadrilateral are A (-2, 4), B (5, 6), C (12, 4) and D (5, 2). Is this a rhombus that is not a square, a rectangle that is not a square, a square, or a trapezoid? B
A C D
All sides are congruent (equal length), but the sides are not perpendicular. Therefore, quadrilateral ABCD is a rhombus.

44. The vertices of a quadrilateral are A (-3, 4), B (4, 4), C (6, -3) and D (-5, -3). Is this a rhombus that is not a square, a rectangle that is not a square, a square, or a trapezoid?
A B
C D
One pair of opposite sides is parallel. Therefore, quadrilateral ABCD is a trapezoid.

45. 11
What type of quadrilateral has vertices (5, 4), (11, 8), (6, 9), and (10, 3)? A
trapezoid
rectangle that is not a square
rhombus that is not a square
square B C D

46. 12
What type of quadrilateral has vertices (4, 7), (3, 1), (-3, 2), and (-2, 8)? A
parallelogram that is not a rectangle/rhombus/square
rectangle that is not a square
rhombus that is not a square
square B C D

47. Midpoint
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48. The midpoint of a segment is the halfway point between the endpoints.8 12 10
midpoint

49. Find the midpoint of the segment.
= 5 = 2.5

50. The midpoint formula (in coordinate plane) finds the average of the x-coordinates and the average of y-coordinates of the endpoints. A B 10 5
A (5, 3) B (9, 5)

51. Find the midpoint of the segment with endpoints A (5,12) and B (-4, 8).,
1 , 20
2 2
(1/2, 10)
ANSWER

52. 10 Endpoint (2, 6) Midpoint (5, 4) Endpoint ( , ) 8 2
Find the missing endpoint C given the midpoint B and the other endpoint A. 10 5
Endpoint (2, 6)
Midpoint (5, 4)
Endpoint ( , ) -2 +3
8 2 A B

53. Endpoint (-1, 4) Midpoint (3, -2) Endpoint ( , ) 7 -8
The midpoint of a line is (3, -2). If one endpoint is (-1, 4), find the other endpoint.
Endpoint (-1, 4)
Midpoint (3, -2)
Endpoint ( , ) +4 -6
7 -8

54. 13
Find the midpoint of the segment with endpoints (5, 2) and (-1, 7).

55. 14
Find the midpoint of the segment with endpoints (-3, -4) and (-2, 2).

56. 15
The midpoint of a line segment is (-3, 9) and one endpoint is (-1, 6). Find the other endpoint.

57. 16
The midpoint of a line segment is (0, 2) and one endpoint is (-6, 1). Find the other endpoint.

58. Area and Volume
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59. Area r d {
Circle
Rectangle
Triangle
Trapezoid b h b h b1 b2 h

60. Find the area of the following figures.
Circle
Rectangle
2 in
6 in
4 cm
A = 42 = cm2
A = 2*6 = 12 in2
Triangle
Trapezoid
15 m
12 m
2 m
3 ft
2 ft
A = 1/2 * 2 * 3 = 3 ft2
A = 1/2*( )*2 = 27 m2

61. The bases of a trapezoid are 3x + 4 and 2x - 3, and the height is 4 inches. Write an expression representing the area of the trapezoid.
2x - 3 4
Answer
3x + 4
A = 1/2 * (2x x + 4) * 4
= 2 * (5x + 1)
= 10x + 2 in2

62. You have enough paint to cover a surface of 20 ft2
You have enough paint to cover a surface of 20 ft2. The rectangular box (rectangular prism) you want to paint is 6 inches tall, 36 inches wide, and 24 inches deep. Do you have enough paint to cover the entire box? If yes, how much more area can you cover?
Top Area
Side Area
Front Area
Bottom Area
Back Area

63. We will have enough to cover the box and an extra 3 ft2.
You have enough paint to cover a surface of 20 ft2. The rectangular box (rectangular prism) you want to paint is 6 inches tall, 36 inches wide, and 24 inches deep. Do you have enough paint to cover the entire box? If yes, how much more area can you cover?
Top Area
Side Area
Front Area
Bottom Area
Back Area
0.5 ft
2 ft
3 ft
height = 6 in = 0.5 feet
width = 36 in = 3 feet
depth = 24 in = 2 feet
Front = Back (3)(0.5) = (1.5) = 3
Top = Bottom (3)(2) = (6) = 12
Side = Side (2)(0.5) = (1) = 2
17 ft2
We will have enough to cover the box and an extra 3 ft2.

64. Volume V = Bh V = Bh V = πr2h Prism Cylinder Pyramid Cone Sphere
Base
height L w h
base
height r
V = Bh
V = πr2h
V = Bh
Square Base (B)
Slant Height (l )
Pyramid's Height (h) r
height
Slant Height l
V = 1/3 Bh
V = 1/3 π r2 h
V = 1/3 Bh r
V = 4/3 π r3

65. Does a prism need to be a right prism for the volume formula to work?
Think of a ream of paper
Stacked nicely it has 500 sheets.
If the stack is fanned, it still has 500 sheets.

66. Example: Find the volume of the box with length 2, width 6, and height 5.

67. Example: The volume of a box is 48 ft3
Example: The volume of a box is 48 ft3. If the height is 4ft and width is 6ft, what is the length.

68. Example: Find the volume of the cylinder with radius 4 and height 11.

69. Example: Find the volume of the pyramid.6 4 5

70. Example: Find the volume of the pyramid.5 8 8

71. Example: Find the volume of the cone.
r= 7 9

72. Example: Find the volume of the cone.12
r= 4

73. Example: Find the volume of the sphere.9

74. Example: Find the volume of the sphere.
Great Circle: A=25π u2

75. 17
What is the volume of a box with edges of 4, 5, and 7?

76. 18
If the volume of a box is 64 u3 and has height 8 and width 4, what is the length?

77. 19
Find the volume of the cylinder with radius 6 and height 8.

78. The height of a cylinder doubles, what happens to the volume?20
The height of a cylinder doubles, what happens to the volume? A
Doubles B
Quadruples C
Depends on the cylinder D
Cannot be determined

79. Find the volume of the pyramid.21
Find the volume of the pyramid. 8 6 6

80. What is the volume of the cone?22
What is the volume of the cone? 8
d=10

81. Find the volume of the sphere.23
Find the volume of the sphere. 4

82. Find the volume of the sphere.24
Find the volume of the sphere.
Great Circle: A= 16π u2