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.

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

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

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

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

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

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

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.

Example

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

Homework Finish worksheet discussed in class

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

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

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 7 900 128.5714286 51.42857143 14 Octagon 8 1080 135 45 20 Nonagon 1260 140 40 27 Decagon 10 1440 144 36 35 Undecagon 11 1620 147.2727273 32.72727273 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

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

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

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

Examples

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

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

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

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

Examples

Examples

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

Homework worksheets

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

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

Examples

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

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

Practice

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

Practice

More Practice

Homework worksheet