1. Chapter 8 Polygons and Circles
Students will use properties of triangles to help determine interior and exterior angles of polygons(review). Students will discovery methods for finding areas of polygons. Students will discovery properties of circles.

2. Section 8.1
Students will discover special intersections points in relationship to triangles.

3. Special segments in Triangles
Medians connects vertex to midpoint of side opposite Height or Altitude segment from vertex, perpendicular to side opposite Angle Bisectors Segment that bisects an angle and touches side opposite Perpendicular Bisector Segment that is the perpendicular bisector of a side Points of Concurrency Where the specific segments above intersect and what is special about them

4. Angle Bisector
Incenter is the point of concurrency – where all three angles bisectors of a triangle intersect, always inside the triangle This point is equidistant from all the sides of the triangle You can use this point as the center of the circle and a point on the each side of the triangle to construct an inscribe circle Used to construct a circle inside the triangle – inscribed circle Inscribed circle – center is incenter, always inside the Triangle and touches all 3 sides, can happen with regular polygons as well

5. Perpendicular Bisector
Circumcenter – this is the point of concurrency, where all 3 perpendicular bisectors intersect Inside triangle is acute On one side if right Outside if obtuse This point is equidistant from all of the vertices If you use the circumcenter as the center of the circle and all the vertices as points on the circle you will construction a circumscribed circle about the triangle Circumscribed circle – center is the circumcenter, always outside the triangle and touches all the vertices, can happen with regular polygons as well

6. Medians
Centriod – is the point of concurrency for the medians of a triangle, always inside the triangle This is the point of the center of gravity for the triangle, could balance the triangle at this point The centroid of a triangle divides each median into 2 parts so that the distance from the centroid to the vertex is twice the distance from the centroid to the midpoint of the opposite side. Centroid to vertex = 2/3 length of median Centroid to side = 1/3 length of median

7. Altitudes
Orthocenter – where the three altitudes intersect Nothing special about this point Inside if acute At right angle if right Outside if obtuse

8. Examples
The first aid station of Mt. Thermopolis State park needs to be located at a point that is equidistant from three bike paths that intersect to form a triangle. Locate this point so that in an emergency, medical personnel will be able to get to any one of the paths by the shortest possible route? Explain your answer.

9. Example

10. Exampe
FP = 5 PC = ? BE= 18 BP = ?

11. Homework
Finish worksheet discussed in class

12. Section 8.2
Students will review interior and exterior angles or polygons and discovery how to find areas of polygons.

13. Angles of Polygons
How do we determine the sum of the interior angles of a polygon 180(n-2) What is the sum of the exterior angles of a polygon 360 How do we find each interior angle of a regular polygon Sum/n How do we find the measure of each exterior angle of a regular polygon 360/n

14. Types and prop of polygons
name
sides
sum of interior angles
measure of each interior angle if regular polygon
sum of exterior angles
measure of each exterior angle if a regular poygon
number of diagonals
Triangles 3
180 60
360
120
Quadrilateral 4 90 2
Pentagon 5
540
108 72
Hexagon 6
720 9
Heptagon 7
900 14
Octagon 8
1080
135 45 20
Nonagon
1260
140 40 27
Decagon 10
1440
144 36 35
Undecagon 11
1620 44
Dodecagon 12
1800
150 30 54
N-agon N
180(n-3)
sum/n
360/n
n(n-3)/2
name
sides
sum of interior angles
measure of each interior angle if regular polygon
sum of exterior angles
measure of each exterior angle if a regular poygon
number of diagonals
Triangles
Quadrilateral
Pentagon
Hexagon
Heptagon
Octagon
Nonagon
Decagon
Undecagon
Dodecagon
N-agon

15. Area
Amount of 2 dimensional space a figure takes up Triangle height is also the altitude (b*h)/2 Rectangle/Parallelograms B*h or L*W Square S^2

16. Trapezoid
Think of this as two triangles – they have different bases Sum of two bases times height divided by 2 Can’t always divide trapezoid into a rectangle and two triangles, the end triangles may not be the same

17. Kite
Split into 2 congruent triangles using the diagonals Area of one triangle times 2

18. Examples

19. Other polygons
How would you find the area of a pentagon Non Regular or Regular Other Shapes

20. Find the area of this regular pentagon, you know the Perimeter is 200 inches.

22. Area
Divide the polygon into Triangles – one vertex is the center of the polygon and the side of the polygon is the base of the triangle – this form congruent isosceles triangles
The base = perim/n
Find the central angle of each isosceles triangle – angle formed at the center
360/n
Draw the altitude of the isosceles triangle to make a right triangle, this bisect the central angle and the base
Calculate the altitude of the triangle using trig ratios Alt= ½ side/ tan (1/2 central angle)
Find the area of the triangle
Find the area of the polygon by multiplying by how many triangles there are

23. Apothem
Apothem is the perpendicular distance from the center of the polygon to one side (height of the triangle) If you know the apothem you can find the area as well a=apothem s=side lengths n=number of sides

24. Examples

25. Examples

26. Similar Polygons and Areas
Remember proportions and scale factor How do we use this to find areas of similar shapes

27. Homework
worksheets

28. Section 8.4
Students will discovery properties of circles and how to calculate area and circumference.

29. Circles
Exact answer or in terms of pi – do not multiply out the pi value Approximate value is when you use, 3.14, 22/7 or the pi button on calculator

30. Examples

31. Parts of Circles - Sector
Chord – segment inside a circle touching two points Sector of a Circle – region between two radii and an arc of the circle, slice of pizza

32. Parts of Circle - Segment
Segment of a circle – is the region between a chord and an arc of a circle, crust of pizza

33. Practice

34. Concentric Circles – same center
different radius
Annulus – the region between
two concentric circles

35. Practice

36. More Practice

37. Homework
worksheet