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# Geometry Smester Project

5 Major Topics

by

Tweet## Tiarra Lowe

on 5 January 2013#### Transcript of Geometry Smester Project

5 Major Topics Geometry Semester Project Midpoint Formula Difference Between Distance and Midpoint Use Midpoint and Distance Formulas Example: Distance Formula * The distance formula tells us the length between two points.

The midpoint of a segment is the point that divides the segment into two congruent segments.

A segment bisector is a point, ray, line or plane that intersects the segment at its midpoint. Key Words to Know Example: M is the midpoint of segment PR.

Line "n" is the segment bisector of segment PR. M = Midpoint ---> Average

A(x1+y1)

B(x2+y2)

M(xm+ym) ---> an average of the two endpoints S(4,-1)

T(6, 0)

4+6 , -1, 10 2 2 = M(5, -1/2) D= A(-3,2)

B(1,5)

D= (-3-1) + (3+3) 2 2 = (-4) + (-3) 2 2 = 16+9 = 25 = 5 Example: If A and B are points in a coordinate

plane, then the distance between

A and B is: * The midpoint formula tells us the center of a segment. Measure and Classify Angles Key Words to Know An angle consists of two different rays with the same endpoint.

The rays are the sides of the angle.

The endpoint is the vertex of the angle Example: Naming Angles Possible names for these angles:

< APC < CPD

< APD < CPB

< APB < DPB Classifying Angles acute angle: < 90 degrees obtuse angle: > 90 degrees right angle: = 90 degrees straight angle: 180 degrees Measuring Angles Use Inductive Reasoning Key Words to Know A conjecture is an unproven statement based on observations.

Inductive reasoning is used to find a pattern in specific cases and then to write a conjecture for the general case.

A counter example is a specific case for which the conjecture is false. Triangle Sum Properties Key Words to Know A triangle is a polygon with three sides. Example: A triangle with vertices A, B, and C is called triangle ABC Classify Triangles by Sides ***How many sides have the same length? ~ Scalene: no equal sides

~ Isosceles: at least two sides are equal

~ Equilateral: all sides are equal Classify Triangles by Angles ***What size angles does the triangle have? ~ Acute: three acute angles

~ Obtuse: one obtuse angle

~ Equiangular: three equal angles

~ Right: one right angle Angles When sides of a polygon are extended, other angles are formed. The original angles are the interior angles.

The angles that form linear pairs with the interior angles are the exterior angles. Classify Polygons

Look carefully at the following figures. Then, use inductive reasoning to make a conjecture about the next figure in the pattern

If you have carefully observed the pattern, may be you came up with the figure below:

Keys Words to Know A polygon is a closed plane figure with the following properties.

It is formed by three or more line segments called sides.

Each endpoint of a side is a vertex of the polygon.

Number of sides

3

4

5

6

7

8

9

10

12 Type of Polygon

triangle

quadrilateral

pentagon

hexagon

heptagon

octagon

nonagon

decagon

dodecagon EXAMPLE

Full transcriptThe midpoint of a segment is the point that divides the segment into two congruent segments.

A segment bisector is a point, ray, line or plane that intersects the segment at its midpoint. Key Words to Know Example: M is the midpoint of segment PR.

Line "n" is the segment bisector of segment PR. M = Midpoint ---> Average

A(x1+y1)

B(x2+y2)

M(xm+ym) ---> an average of the two endpoints S(4,-1)

T(6, 0)

4+6 , -1, 10 2 2 = M(5, -1/2) D= A(-3,2)

B(1,5)

D= (-3-1) + (3+3) 2 2 = (-4) + (-3) 2 2 = 16+9 = 25 = 5 Example: If A and B are points in a coordinate

plane, then the distance between

A and B is: * The midpoint formula tells us the center of a segment. Measure and Classify Angles Key Words to Know An angle consists of two different rays with the same endpoint.

The rays are the sides of the angle.

The endpoint is the vertex of the angle Example: Naming Angles Possible names for these angles:

< APC < CPD

< APD < CPB

< APB < DPB Classifying Angles acute angle: < 90 degrees obtuse angle: > 90 degrees right angle: = 90 degrees straight angle: 180 degrees Measuring Angles Use Inductive Reasoning Key Words to Know A conjecture is an unproven statement based on observations.

Inductive reasoning is used to find a pattern in specific cases and then to write a conjecture for the general case.

A counter example is a specific case for which the conjecture is false. Triangle Sum Properties Key Words to Know A triangle is a polygon with three sides. Example: A triangle with vertices A, B, and C is called triangle ABC Classify Triangles by Sides ***How many sides have the same length? ~ Scalene: no equal sides

~ Isosceles: at least two sides are equal

~ Equilateral: all sides are equal Classify Triangles by Angles ***What size angles does the triangle have? ~ Acute: three acute angles

~ Obtuse: one obtuse angle

~ Equiangular: three equal angles

~ Right: one right angle Angles When sides of a polygon are extended, other angles are formed. The original angles are the interior angles.

The angles that form linear pairs with the interior angles are the exterior angles. Classify Polygons

Look carefully at the following figures. Then, use inductive reasoning to make a conjecture about the next figure in the pattern

If you have carefully observed the pattern, may be you came up with the figure below:

Keys Words to Know A polygon is a closed plane figure with the following properties.

It is formed by three or more line segments called sides.

Each endpoint of a side is a vertex of the polygon.

Number of sides

3

4

5

6

7

8

9

10

12 Type of Polygon

triangle

quadrilateral

pentagon

hexagon

heptagon

octagon

nonagon

decagon

dodecagon EXAMPLE