Transcript Chapter 5 Perimeter and Area

Chapter 5
Perimeter and Area
5-1: Perimeter and Area
5-2: Areas of Triangles, Parallelograms, and Trapezoids
5-3: Circumferences and Areas of Circles
5-4:The Pythagorean Theorem
5-5: Special Triangles and Areas of Regular Polygons
5-6: The Distance Formula and the Method of Quadrature
5-7: Proofs Using Coordinate Geometry
5-8: Geometric Probability
5.1 Perimeter and Area
•
•
Perimeter—the distance around an
object
To find the perimeter of a polygon, add
the lengths of all of its sides.
Area—the surface encompassed
by a polygon
Area of Triangle=1/2bh, where height, h, is
perpendicular to the base
Area of Square=s2 where s is the
length of the side
Area of Rectangle = l x w, where l
is the length and w is the width
5.2 Areas of Triangles,
Parallelograms, and Trapezoids
• Area of Triangle=1/2bh, where h is
perpendicular to the base, b; h does not
have to touch the base
Area of Parallelogram=b x h, where
height, h, is perpendicular to base, b
Area of Trapezoid=1/2(h)(b1+b2) or
A=h(midsegment); h is height
perpendicular to and intersecting
both bases (b1and b2)
5.3 Circumferences and Areas of
Circles
• Circumference—perimeter of a circle
• C=2πr or C=2d, where r is the radius and
d is the diameter
Area of Circle =πr2
If they give you the area or
circumference and ask for the
radius, you must use algebra to
solve for r.
5.4 The Pythagorean Theorem
• Pythagorean Theorem: Given a right triangle,
the sum of the squares of the legs is equal to the
square of the hypotenuse
• hyp2=leg2+leg2 or c2=a2+b2
•
Your longest leg is always c!
Pythagorean Triples: Sets of
lengths that always make a right
triangle
3,4,5
5,12, 13
7,24,25
9, 40, 41
11, 60, 61
Using Side Lengths to Determine if a
Triangle is Right, Acute, or Obtuse
Converse of the Pythagorean Theorem:
If c2=a2+b2, then you have a right triangle.
If c2<a2+b2, then you have an acute triangle.
If c2>a2+b2, then you have an obtuse
triangle.
5.5 Special Triangles and Areas of
Regular Polygons
• Special Triangle 1: 30-60-90
•Take an equilateral triangle with
sides of length 2a and split it in
half. This leaves you with a 3060-90 triangle.
•Since everything was split in half,
the base of this 30-60-90 triangle
is half of the original, or a.
•Use the Pythagorean
Theorem to get the height:
Special Triangle 1: 30-60-90
• So, in a 30°-60°-90° triangle, the side
opposite 30° is a, the side opposite 60° is
a√3, and the side opposite 90° is 2a
a√3
•When solving problems with
a 30-60-90 triangle, set the
known side length equal to its
ratio.
•Solve for a.
•Then substitute into the other
ratios to find the missing side
lengths.
Special Triangle 2: 45-45-90
• Take a 45-45-90 triangle with
sides of length a. (Since the
angles are congruent, this
triangle is isosceles.
• Use the Pythagorean
Theorem to find the length of
the hypotenuse.
Special Triangle 2: 45-45-90
So, in a 45°-45°-90° triangle, the sides
opposite 45° are a, and the side opposite
90° is a√2
•When solving problems with
a 45-45-90 triangle, set the
known side length equal to its
ratio.
•Solve for a.
•Then substitute into the other
ratios to find the missing side
lengths.
Finding the Area of a Regular
Polygon
• Let’s say we have a hexagon with side length 10.
1. Put the hexagon into a circle. How
many triangles do we get? What
are the angles inside?
2. Draw the height of the triangles
(from the vertex of the circle to the
middle of the edge). This is called
your apothem in a regular polygon.
Finding the Area of a Regular
Polygon
3. Use special triangles to find the length of
the apothem and calculate the area of
the triangle.
4. Multiply this area by the number of
triangles in the hexagon to get the total
area.
Our side length=10, so
a=5, and the height
must be 5√3.
Finding the Area of a Regular
Polygon
Formula for the area of a regular polygon:
• A=1/2(apothem)(perimeter)
• p=(number of sides)(length of side)
5.6 The Distance Formula and the
Method of Quadrature
• Distance—the length between two points
• Given points (x1, y1) and (x2, y2)
Choose point one and point two and
substitute in this formula to find
distance.
5.7 Proofs Using Coordinate
Geometry
• Midpoint—the point which divides a segment
into two congruent parts
• Given endpoints (x1, y1) and (x2, y2)
Slope=
• Use properties of polygons to find determine
lengths of sides and points of vertices.
5.8 Geometric Probability
• Probability is the likelihood that an event will
happen. It is always between 0 and 1. (0
means that it is impossible, and 1 means that it
always has to happen.)
• P= (#favorable outcomes)
(#possible outcomes)
• In geometry, probability is usually the percent of
area an “event” represents—like the area
represented by blue on a spin dial.