9.2 – Curves, Polygons, and Circles

9.2 – Curves, Polygons, and CirclesCurvesThe basic undefined term curve is used for describing non-linear figures in a plane.A simple curve can be drawn without lifting the pencil from the paper, and without passing through any point twice.A closed curve has the same starting and ending points, and is also drawn without lifting the pencil from the paper.

Simple; closed

Simple; not closed

Not simple; closed

Not simple; not closed

9.2 – Curves, Polygons, and Circles

A polygon is a simple, closed curve made up of only straight line segments.

Polygons with all sides equal and all angles equal are regular polygons.

The line segments are called sides. The points at which the sides meet are called vertices.

Polygons

Regular PolygonsPolygons

9.2 – Curves, Polygons, and CirclesA figure is said to be convex if, for any two points A and B inside the figure, the line segment AB is always completely inside the figure.

AE F

Convex Not convex

B

DC

M N

9.2 – Curves, Polygons, and CirclesClassification of Polygons According to Number

of Sides

Number of Sides

Name Number of Sides

Names

3 Triangle 13

4 Quadrilateral 14

5 Pentagon 15

6 Hexagon 16

7 Heptagon 17

8 Octagon 18

9 Nonagon 19

10 Decagon 20

11 30

12 40

Hendecagon

Dodecagon

Tridecagon

Tetradecagon

Pentadecagon

Hexadecagon

Heptadecagon

Octadecagon

Nonadecagon

Icosagon

Triacontagon

Tetracontagon


Types of Triangles - Angles

All Acute Angles One Right Angle One Obtuse Angle

Acute Triangle Right Triangle Obtuse Triangle


Types of Triangles - Sides

All Sides Equal Two Sides Equal No Sides Equal

Equilateral Triangle Isosceles Triangle

Scalene Triangle

9.2 – Curves, Polygons, and CirclesQuadrilaterals: any simple and closed four-sided figure

A rectangle is a parallelogram with a right angle.

A trapezoid is a quadrilateral with one pair of parallel sides.

A parallelogram is a quadrilateral with two pairs of parallel sides.

A square is a rectangle with all sides having equal length.

A rhombus is a parallelogram with all sides having equal length.


The sum of the measures of the angles of any triangle is 180°.Angle Sum of a Triangle

Triangles

Find the measure of each angle in the triangle below.

(x+20)°

x°(220 – 3x)°

x + x + 20 + 220 – 3x = 180–x + 240 = 180

– x = – 60 x = 60

x = 60°60 + 20 = 80°

220 – 3(60) = 40°

9.2 – Curves, Polygons, and CirclesExterior Angle Measure

The measure of an exterior angle of a triangle is equal to the sum of the measures of the two opposite interior angles.

1

2

34

The measure of angle 4 is equal to the sum of the measures of angles 2 and 3.

m4 = m2 + m3

Exterior angle

9.2 – Curves, Polygons, and CirclesFind the measure of the exterior indicated below.

3x – 50 = x + x + 20

(x+20)°

x°(3x – 50)° (x – 50)°

3x – 50 = 2x + 20

x = 703x = 2x + 70

3(70) – 50160°

9.2 – Curves, Polygons, and CirclesCircles

A circle is a set of points in a plane, each of which is the same distance from a fixed point (called the center).A segment with an endpoint at the center and an endpoint on the circle is called a radius (plural: radii).A segment with endpoints on the circle is called a chord.A segment passing through the center, with endpoints on the circle, is called a diameter.

A line that touches a circle in only one point is called a tangent to the circle. A line that intersects a circle in two points is called a secant line.

A portion of the circumference of a circle between any two points on the circle is called an arc.

A diameter divides a circle into two equal semicircles.


P

R

O

T

Q

RT is a tangent line.

PQ is a secant line.

OQ is a radius.

(PQ is a chord).

O is the center

PR is a diameter.

PQ is an arc.M

L

LM is a chord.

9.2 – Curves, Polygons, and CirclesInscribed Angle

Any angle inscribed in a semicircle must be a right angle.

An inscribed angle is an angle whose vertex is on the circle and the sides of the angle extend through the circle and touch or go beyond the points on the circle.

A

B

C

x

BAC is an inscribed angle

A

B

C

9.3 –Triangles: Congruence, Similarity and the Pythagorean Theorem

Congruent Triangles

A

B

C

Congruent triangles: Triangles that are both the same size and same shape.

D

E

FThe corresponding sides are congruent.

.ABC DEF

The corresponding angles have equal measures.

Notation:

Using Congruence Properties Side-Angle-Side (SAS) If two sides and the included angle of one triangle are equal, respectively, to two sides and the included angle of a second triangle, then the triangles are congruent.

A

B

CD

E

F

.ABC DEF


Using Congruence PropertiesAngle-Side-Angle (ASA) If two angles and the included side of one triangle are equal, respectively, to two angles and the included side of a second triangle, then the triangles are congruent.

A

B

CD

E

F

.ABC DEF


A

B

CD

E

F

Congruence Properties - SSS

Side-Side-Side (SSS) If three sides of one triangle are equal, respectively, to three sides of a second triangle, then the triangles are congruent.

.ABC DEF


Proving Congruence Given:

A

B

DE

C

Given

Prove:

CE = ED

AE = EB

ACE BDE

STATEMENTS REASONS

CE = ED

AE = EB Given

ACE BDE

AEC = BED Vertical angles are equal

SAS property

1. 1.

2. 2.

3. 3.

4. 4.



Given

Prove:

ADB = CBD

ABD CDB

STATEMENTS REASONS

Given

Reflexive property

ASA property

1. 1.

2. 2.

3. 3.

4. 4.

ABD = CDB

ADB = CBD

ABD = CDB

ABD CBD

A

B

D

C

BD = BD



Given

Prove: ABD CBD

STATEMENTS REASONS

Given

Reflexive property

SSS property

1. 1.

2. 2.

3. 3.

4. 4.ABD CBD

A

B

D C

AD = CD

AB = CB

AD = CD

AB = CB

BD = BD


Isosceles Triangles If ∆ABC is an isosceles triangle with AB = CB, and if D is the midpoint of the base AC, then the following properties hold.

1. The base angles A and C are equal.2. Angles ABD and CBD are equal.3. Angles ADB and CDB are both right angles.

A C

B

D


Similar Triangles: Triangles that are exactly the same shape, but not necessarily the same size.

For triangles to be similar, the following conditions must hold:1. Corresponding angles must have the same measure.2. The ratios of the corresponding sides must be constant. That is, the corresponding sides are proportional.

If the measures of two angles of one triangle are equal to those of two corresponding angles of a second triangle, then the two triangles are similar.

Angle-Angle Similarity Property


is similar to .ABC DEF

Find the length of side DF.

A

B

CD

E

F16

24

32

8

Set up a proportion with corresponding sides:

EF DFBC AC

8

16 32DF

DF = 16.


Pythagorean Theorem

If the two legs of a right triangle have lengths a and b, and the hypotenuse has length c, then

2 2 2.a b c

leg b

leg ac (hypotenuse)

(The sum of the squares of the lengths of the legs is equal to the square of the hypotenuse.)


Find the length a in the right triangle below.

2 2 2a b c

39

36

a

2 2 236 39a 2 1296 1521a

2 225a 15a


Converse of the Pythagorean Theorem If the sides of lengths a, b, and c, where c is the length of the longest side, and if 2 2 2 ,a b c then the triangle is a right triangle.

Is a triangle with sides of length 4, 7, and 8, a right triangle?

2 2 24 7 8 16 49 64

65 64

Not a right triangle.


Is a triangle with sides of length 8, 15, and 17, a right triangle?

right triangle.

9.4 – Perimeter, Area, and CircumferencePerimeter of a Polygon

The perimeter of any polygon is the sum of the measures of the line segments that form its sides. Perimeter is measured in linear units.Perimeter of a Triangle

a b

cThe perimeter P of a triangle with sides of lengths a, b, and c is given by the formula:

P = a + b + c.

9.4 – Perimeter, Area, and Circumference

Perimeter of a Rectangle wl

The perimeter P of a rectangle with length l and width w is given by the formula:

P = 2l + 2w or P = 2(l + w).

Perimeter of a Square

The perimeter P of a square with all sides of length s is given by the formula:

ss

P = 4s.

9.4 – Perimeter, Area, and CircumferenceArea of a Polygon

The amount of plane surface covered by a polygon is called its area. Area is measured in square units.

Area of a Rectangle

The area A of a rectangle with length l and width w is given by the formula:

wl

A = l w.

Area of a Square ss

The area A of a square with all sides of length s is given by the formula:

P = s2.


Area of a Parallelogram

The area A of a parallelogram with height h and base b is given by the formula:

b

h

A = bh.

Area of a Trapezoid

The area A of a trapezoid with parallel bases b1 and b2 and height h is given by the formula:

b2

h

b1

A = (1/2) h (b1 + b2)


Area of a Triangle

The area A of a triangle with base b and height h is given by the formula:

h

b

A = (1/2) b h

9.4 – Perimeter, Area, and CircumferenceFind the perimeter and area of the rectangle.

7 ft15 ft

P = 44

Perimeter

Area A = 105

P = 2l + 2w 2(15) + 2(7)

A = lw

Find the area of the trapezoid.

13 cm.

5 cm.

7 cm.

A = (1/2) h (b1 + b2)

A = (1/2) (5) (7 + 13)

A = (1/2) (5) (20)

A = 50 cm2

6 cm. 6 cm.

ft2

ft

A = (15)(7)

9.4 – Perimeter, Area, and CircumferenceFind the area of the shaded region.

Area of square 4 in.

4 in.

– Area of triangle

s2 – (1/2) b h

42 – (1/2) (4)(4)

16 – 8 8 in2

9.4 – Perimeter, Area, and CircumferenceCircumference and Area of a Circle

The circumference C of a circle of diameter d is given by the formula: or where r is a radius.,C d 2 ,C r

d r

The area A of a circle with radius r is given by the formula:2.A r

9.4 – Perimeter, Area, and CircumferenceFind the area and circumference of a circle with a radius that is 6 inches long (use 3.14 as an approximation for ).

Circumference ( 3.14)

Area ( 3.14)

C = 2 rC = 2 (3.14) 6C = 37.68 in

A = r2

A = (3.14) 62

A = 113.04 in2

Circumference ()C = 2 rC = 2 6

C = 37.699 in

Area ()A = r2

A = 62

A = 113.097 in2

9.5 – Space Figures, Volume, and Surface Area

9-437

Space figures: Figures requiring three dimensions to represent the figure.Polyhedra: Three dimensional figures whose faces are made only of polygons.Regular Polyhedra: A polyhedra whose faces are made only of regular polygons (all sides are equal and all angles are equal.

9.5 – Space Figures, Volume, and Surface AreaOther Polyhedra

Pyramids are made of triangular sides and a polygonal base. Prisms have two faces in parallel planes; these faces are congruent polygons.

9.5 – Space Figures, Volume, and Surface AreaOther Space Figures

Right Circular Cylinders have two circles as bases, parallel to each other and whose centers are directly above each other.

Right Circular Cones have a circle as a base and the surface tapers to a point directly above the center of the base.

9.5 – Space Figures, Volume, and Surface AreaVolume and Surface Area

Volume is a measure of capacity of a space figure. It is always measured in cubic units.Surface Area is the total region bound by two dimensions. It is always measured in square units.

w

h

l

The volume V and surface area S of a box with length l, width w, and height h is given by the formulas:

V = lwh

Volume and Surface Area of a Rectangular solid (box)

andS = 2lw + 2lh +

2hw


9-441

2 in.

3 in.

7 in.

Find the volume and surface area of the box below.

V = 7(2)(3)

V = 42 in.3

S = 2(7)(2) + 2(7)(3) + 2(3)(2)

S = 28 + 42 + 12

S = 82 in.2


The volume V and surface area S of a cube with side lengths of s are given by the formulas:

V = s3

Volume and Surface Area of a Cube

s

ss

5 ft.

Find the volume and surface area of the cube below.

V = 53 S = 6(5)2

V = 125 ft.3 S = 625

S = 150 ft.2

andS = 6s2


Volume of Surface Area of a Right Circular Cylinder

10 m2 m

The volume V and surface area S of a right circular cylinder with base radius r and height h are given by the formulas:

Volume and Surface Area

Find the volume and surface area of the cylinder below.V = (2)2(10)V = 40V = 125.6 m3

S = 2(2)(10) + 2(2)2 S = 40 + 8 = 48 S = 150.72 m2

V = r2hh

r

andS = 2rh + 2r2


Volume of Surface Area of a Sphere

r

The volume V and surface area S of a sphere radius r are given by the formulas:

Volume and Surface Area

V = (4/3) r3

Find the volume and surface area of the sphere below.

9 in.S = 4 (9)2V = (4/3)(9)3

V = 972V = 3052.08 in.3

S = 324S = 1017.36 in.2

andS = 4 r2


Volume of Surface Area of a Right Circular Cone The volume V and surface area S of a right circular cone with base radius r and height h are given by the formulas:

9.5 – Space Figures, Volume, and Surface AreaFind the volume and surface area of the cone below.

V = (1/3)(3)2(4)V = 12V = 37.68 m3

S = 15 + 9S = 24S = 75.36 m2

h = 4 m

r = 3 m


Volume of a Pyramid The volume V of a pyramid with height h and base of area B is given by the formula:

Find the volume of the pyramid (rectangular base) below.

cm3

Note: B represents the area of the base (l w).

7 cm

6 cm3 cm

9.2 – Curves, Polygons, and Circles

Documents

Transcript of 9.2 – Curves, Polygons, and Circles