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Section 5.1. Perimeter and Area. Perimeter and Area . The perimeter of a closed plane figure is the distance around the figure. The area of a closed plane figure is the number of non-overlapping squares of a given size that will exactly cover the interior of the figure. - PowerPoint PPT Presentation
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Section 5.1
Perimeter and Area
Perimeter and Area
• The perimeter of a closed plane figure is the distance around the figure.
• The area of a closed plane figure is the number of non-overlapping squares of a given size that will exactly cover the interior of the figure.
Finding the Perimeter
• Hexagon ABCDEF Quadrilateral WXYZ
B 78 C
5 W 12 A D X 10 13 12 7 F E Z Y 4 5
AB + BC + CD + DE + EF + FA = Perimeter WX + XY + YZ + ZW = Perimeter
Finding the Perimeter of Rectangles and Squares
• The perimeter of a rectangle with base b and height h is given by: P = 2b + 2h.
b s
h h s s
b s
The perimeter of a square with a side s is given by: P = 4s.
Finding the Perimeter of Rectangles and Squares
Find the perimeter of the rectangle and square.
8 m
20 m 5 in
P = 2(8) + 2(20) P = 4(5)P = 16 + 40 P = 20 inP = 56 m
Find the Area of Rectangles and Squares
• The area of a rectangle with base b and height h is given by: A = bh. (length x width)
h s
b
The area of a square with a side s is given by: P = s².
Find the Area of Rectangles and Squares
• Find the area of the rectangle and square.
12 ft 7 mi.
23 ft
A = 12(23) A = 7²A = 276 ft² A = 49 mi²
Section 5.2
Areas, of Triangles, Parallelograms, and Trapezoids
Parts of Triangles
• Any side of a triangle can be called the base of the triangle. The altitude of the triangle is a perpendicular segment from a vertex to a line containing the base of the triangle. The height of the triangle is the length of the altitude.
Altitude
Base
Area of a Triangle
• For a triangle with base b and height h, the area, A, is given by: A = ½bh.
G T 35 in.
29 in. 21 in. 50 mi. 30 mi. L
H 48 in. S 25 mi. RA = ½(48)(21) A = ½(25)(30)A = 504 in.² A = 375 mi.²The area of ∆GHL is 504 in.² The area of ∆RTS is 504 mi.²
Parts of a Parallelogram
• Any side of a parallelogram can be called the base of the parallelogram. An altitude of a parallelogram is a perpendicular segment from a line containing the base to a line containing the side opposite base. The height of the parallelogram is the length of the altitude.
Altitude
Base
Area of a Parallelogram
• For a parallelogram with base b and height h, the area, A, is given by: A = bh
MC D N 9
5 ft. 7 ft. 8 F E P
13 ft O 4A = (13)(5) A = (4)(8)A = 65 ft.² A = 32 un.²The area of CDEF is 65 ft.² The area of NOPM is 32 un.²
Parts of a Trapezoid• The two parallel sides of a trapezoid are known as the
bases of the trapezoid. The two nonparallel sides are called the legs of the trapezoid. An altitude of a trapezoid is a perpendicular segment from a line containing one base to a line containing the other base. The height of a trapezoid is the length of an altitude. Base (b )₁ Leg Leg
Altitude Base (b )₂
Area of a Trapezoid
• For a trapezoid with bases b ₁ and b₂, and height h, the area, A, is given by:
• A = ½(b₁ + b₂)h 50 ft. X Y A = ½(50 + 30)23
23 ft. 32 ft. A = ½(80)23A = 920 ft.²
W 30 ft. Z The area of trapezoid WXYZ is 920 ft.²
Section 5.3
Circumferences and Areas of Circles
Definition of a Circle
• A circle is the set of all points in a plane that are the same distance, r, from a given point in the plane known as the center of the circle. The distance r is known as the radius of the circle. The distance d = 2r is known as the diameter.
• The radius is a segment whose endpoints are located in the center of the circle and on the circle.
• The diameter is a segment whose endpoints are both located on the circle and must pass through the center of the circle.
Diagram of a Circle
Diameter = dRadius = r
d Center
r
Circumference of a Circle
• The circumference, C, of a circle with diameter d and radius r is given by:
• C = πd or C = 2πr
7 20
C = 2π(7) C = π20C = 43.98 un. (approx. answer) C = 62. 83 un. (approx. answer)C = 14π (exact answer) C = 20π (exact answer)
Area of a Circle
• The area, A, of a circle with radius r, is given by: A = πr²
3 8
A = π(3²) A = π(4²)A = 9π A = 16πA = 28.27 un² A = 50.27 un²
Section 5.4
The Pythagorean Theorem
Pythagorean Theorem
• For any right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.
• a² + b² = c²• (leg )₁ ² + (leg )₂ ² = (hyp)²
a c (leg )₁ (hyp)b (leg )₂
Pythagorean Theorem B C 9 12 y 8
AM x R 15
D
(9²) + (12²) = x² (8²) + (15²) = y²81 + 144 = x² 64 + 225 = y²225 = x² 289 = y²√(225) = √(x²) √(289) = √(y²)15 = x 17 = y
The Converse of the Pythagorean Theorem
• If the square of the length of one side of a triangle equals the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle.
G w² + 24² = 25² w² + 576 = 625
25 - 576 - 576 w² = 49
H L √(w²) = √(49)24 w = 7
Pythagorean Inequalities
• For ∆ABC, with c as the length of the longest side:
• If c² = a² + b², then ∆ABC is a right triangle.• If c² > a² + b², then ∆ABC is an obtuse triangle.• If c² < a² + b², then ∆ABC is an acute triangle.
B a c C b A
Radical Form
If a right triangle has leg measurements of 7 and 8, what is the length of the hypotenuse?
7² + 11² = x² Radical Form49 + 121 = x² √(160) = √(2 5∙ ∙4 4∙ ) Prime
#’s160 = x² √(160) = 4√(2 5) Group∙√(160) = √(x²) √(160) = 4√(10)12.65 ≈ x (approximate answer) x = 4√(10) (radical
form)(exact
answer)
Section 5.5
Special Triangles and Areas of Regular Polygons
45-45-90 Triangle Theorem
• In any 45-45-90 triangle, the length of the hypotenuse is √(2) times the length of a leg.
A H n 45⁰ n√(2) 10 45⁰ 10√(2)
45⁰ 45⁰ C n B G 10 F
30-60-90 Triangle Theorem
• In any 30-60-90 triangle, the length of the hypotenuse is 2 times the length of the shorter leg, and the length of the longer leg is √(3) times the length of the shorter leg.
T M 60⁰ 60⁰
x 2x 5 2(5) 30⁰ 30⁰
R x√(3) S O 5√(3) N
Area of a Regular Polygon
• The area, A, of a regular polygon with apothem a and perimeter P is given by: A = ½ap
• An altitude of a triangle from the center of the polygon to the center of a side of the polygon is called an apothem of the polygon.
Area of a Regular Polygon
• A regular polygon is a polygon that has all of its sides congruent and all of its angles congruent.
120⁰ 120⁰
120⁰ 120⁰ apothem
120⁰ 120⁰
Area of a Regular Polygon
A A = ½aP P = 5(7) 7 7 A = ½(4)(35) 4 A = ½(140) B E A = 70 units² 7 7 C 7 D
Section 5.6
The Distance Formula and the Method of Quadrature
Distance Formula
• In a coordinate plan, the distance, d, between two points (x , y ) and ₁ ₁ (x₂, y₂) is given by the following formula: d = √((x₂ - y₂)² + (x₁ - y₁)²)
• Find the distance between A (3, 4) and B(2, 7).d = √((2 – 3)² + (7 – 4)²)d = √((- 1)² + (3)²)d = √(1 + 9)d = √(10) (exact answer)d ≈ 3.16 (approximate answer)
Distance Formula
• Determine if the following three coordinates given could be used for a right triangle.
• (2, 1), (6, 4), (- 4, 9)d = √((6 – 2)² + (4 – 1)²) d = √((- 4 – 2)² + (9 – 1)²) d = √((- 4 – 6)² + (9 –
4)²)d = √((4)² + (3)²) d = √((- 6)² + (8)²) d = √((- 10)² + (5)²)d = √(16 + 9) d = √(36 + 64) d = √(100 + 25)d = √(25) d = √(100) d = √(125)d = 5 d = 10 d = 5√(5) or ≈ 11.185² + 10² = √(125)² (Pythagorean Theorem)25 + 100 = 125 Since 25 + 100 = 125, then yes these are the coordinates
of a right triangle.
The Method of Quadrature
• The area of an enclosed region on a plane can be approximated by the sum of the areas of a number of rectangles. The technique, called quadrature, is particularly important for finding the area under a curve.
Section 5.7
Proofs Using Coordinate Geometry
Midpoint Formula
• The midpoint of a segment with endpoints (x₁, y₁) and (x₂, y₂) has the following coordinates: x + x , y + y₁ ₂ ₁ ₂ ͞͞͞͞͞͞͞͞͞͞͞͞͞͞͞͞͞͞͞ ͞ ͞ ͞ ͞ ͞ ͞ ͞͞ ͞ ͞ ͞ ͞ ͞ ͞ ² ²
The Triangle Midsegment Theorem
Vertices of
triangles
Coordinates of midpoint
M
Coordinates of midpoint
SSlope Slope Length Length
AB BC MS AC MS AC
A(0, 0), B(2, 6), C(8, 0)
M(1, 3) S(5, 3) 0 0 4 8
A(0, 0), B(6, -8),C(10, 0)
M(3, - 4) S(8, - 4) 0 0 5 10
A(0, 0),B(2p, 2q),
C(2r, 0)M(p, q) S(p + r, q) 0 0 r 2r
The Diagonals of a Parallelogram
Three vertices of a parallelogram Fourth vertex Midpoint of BD Midpoint of AC
A(0, 0), B(2, 6), D(10, 0) C(12, 6) (6, 3) (6, 3)
A(0, 0), B(2p, 2q), D(2r, 0) C(2p + 2r, 2q) (p + r, q) (p + r, q)
Section 5.8
Geometric Probability
Probability
• Probability is a number from 0 to 1 (or from 0 to 100 percent) that indicates how likely an event is to occur.
• A probability of 0 (or 0 percent) indicates that the event cannot occur.
• A probability of 1 (or 100 percent) indicates that the event will definitely occur.
Theoretical Probability
• For many situations, it is possible to define and calculate the theoretical probability of an event with mathematical precision. The theoretical probability that an event will occur is a fraction whose denominator represents all equally likely outcomes and whose numerator represents the outcomes in which the event occurs.
Experimental Probability
• In experimental probability, the activity is actually performed with a number of trials. During the experiment, results are recorded then the probability of an outcome is calculated. The numerator includes the number of favorable outcomes and the denominator includes the total number of trials that took place during the experiment.
