Chapter 16

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# Chapter 16 - PowerPoint PPT Presentation

Chapter 16. Geometry of three-dimensional solids. Axiomatic systems. TOK. Geometry was developed along axiomatic systems. This is where everything has to work within a set of rules. Three-point geometry has these rules: There exist exactly three different points.

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Presentation Transcript

### Chapter 16

Geometry of three-dimensional solids

TOK

Geometry was developed along axiomatic systems. This is where everything has to work within a set of rules.

Three-point geometry has these rules:

• There exist exactly three different points.
• Each pair of points lies on exactly one line.
• Each two distinct lines are on at least one point.

What would the axioms be for four-point geometry?

TOK

The angles at the corners of a

That’s a fact.

Isn’t it?

This triangle is drawn on a sphere.

Do its corners add up to 180°?

Pyramid

To work out the height of the triangle face:

Drop a vertical line down from the vertex (top corner)

to the centre of the base, then join the centre of the base

to the midpoint of an edge of the base.

Pyramid

To work out the length of a diagonal edge from base to vertex:

Drop a vertical line from the vertex down to the centre of the base, and join the centre of the base to a corner of the base.

Cuboid

To work out the length of the diagonal across one rectangular face:

Draw from the ends of the diagonal to either of the untouched corners of the rectangle.

Cuboid

To work out the length of the

three-dimensional diagonal across the box:

From one end, draw along the edge of the box to the next corner. From the other end, go diagonally across the appropriate face to meet the first line.

Cone

To work out the slant height of the cone:

Drop a vertical line down from the vertex to the centre of the base, and join the centre of the base to the curved edge of the base.

When a line is on a plane, it merges with the plane.

Imagine this line being rotated up off the plane.

This creates a line in a different plane

and thus needs three dimensions.

The line should meet the plane or go through it.

Imagine the shadow of the line on the plane.

Draw a perpendicular line from the top of the line to the plane,

so that it meets the shadow of the line.

Use this triangle to work out the angle between the line and the plane.

The line should meet the plane or go through it.

Imagine the shadow of the line on the plane.

Draw a vertical line down from the top of the line to the shadow on the plane to make a right angle.

Use this triangle to work out the angle between the line and the plane.