# Euclid: A legend in mathematics - PowerPoint PPT Presentation

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Euclid: A legend in mathematics. "Hitch your wagon to a star." --Ralph Waldo Emerson. The father of Geometry. Euclid developed his mathematical reasoning based on POSTULATES or what some people call AXIOMS (say it like “axe-e- yums ”).

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Euclid: A legend in mathematics

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## Euclid: A legend in mathematics

"Hitch your wagon to a star."--Ralph Waldo Emerson

### The father of Geometry

• Euclid developed his mathematical reasoning based on POSTULATES or what some people call AXIOMS (say it like “axe-e-yums”)

• Postulates or what some people call axioms are used to prove THEOREMS (say it like “thear-ums” where “thear” rhymes with fear).

In case you thought I forgot!

• Postulates or Axioms: are accepted rules without proof.

• Theorems: are rules that can be proven.

### Points

• Points do not have actual size.

• How to Sketch:

Using dots

• How to label:

Use capital letters

Never name two points with the same letter

(in the same sketch).

A

B

A

C

### Lines

• Lines extend indefinitely and have no thickness or width.

• How to sketch : using arrows at both ends.

• How to name: 2 ways

(1) small script letter – line n

(2) any two points on the line -

• Never name a line using three points -

n

A

B

C

### Collinear Points

• Collinear points are points that lie on the same line. (The line does not have to be visible.)

• A point lies on the line if the coordinates of the point satisfy the equation of the line.

Ex: To find if A (1, 0) is collinear with

the points on the line y = -3x + 3.

Substitute x = 1 and y = 0 in the equation.

0 = -3 (1) + 3

0 = -3 + 3

0 = 0

The point A satisfies the equation, therefore the point is collinear

with the points on the line.

A

B

C

Collinear

C

A

B

Non collinear

### Planes

• A plane is a flat surface that extends indefinitely in all directions.

• How to sketch: Use a parallelogram (four sided figure)

• How to name: 2 ways

(1) Capital script letter – Plane M

(2) Any 3 non collinear points in the plane - Plane: ABC/ ACB / BAC / BCA / CAB / CBA

A

M

B

C

Horizontal Plane

Vertical Plane

Other

A

B

Plane ABCD

Plane EFGH

Plane BCGF

Plane ABFE

Plane CDHG

Etc.

D

C

E

F

H

G

### Other planes in the same figure:

Any three non collinear points determine a plane!

Plane AFGD

Plane ACGE

Plane ACH

Plane AGF

Plane BDG

Etc.

### Coplanar Objects

Coplanar objects (points, lines, etc.) are objects that lie on the same plane. The plane does not have to be visible.

Are the following points coplanar?

A, B, C ?

Yes

A, B, C, F ?

No

H, G, F, E ?

Yes

E, H, C, B ?

Yes

A, G, F ?

Yes

C, B, F, H ?

No

### Intersection of Figures

The intersection of two figures is the set of points that are common in both figures.

The intersection of two lines is a point.

m

Line m and line n intersect at point P.

P

n

Continued…….

### 3 Possibilities of Intersection of a Line and a Plane

(1) Line passes through plane – intersection is a point.

(2) Line lies on the plane - intersection is a line.

(3) Line is parallel to the plane - no common points.

### Intersection of Two Planes is a Line.

B

P

A

R

Plane P and Plane R intersect at the line