E5.1 - Planes and further solids (Extended)

Planes and bisectors

Suppose a line meets a $10\,\mathrm{cm}$ segment at a right angle but cuts it into lengths $4\,\mathrm{cm}$ and $6\,\mathrm{cm}$. It is perpendicular to the segment, but what is missing from the name “perpendicular bisector”?

A tabletop suggests part of a plane, but a mathematical plane has no edge. In a sketch, a sloping parallelogram often represents a finite portion of a plane. The perspective does not mean that the surface is curved or thick.

The word bisect means divide into two equal parts. If $M$ is the midpoint of $AB$, then $AM=MB$. A perpendicular bisector must satisfy two separate conditions: pass through $M$, and meet $AB$ at a right angle.

[DIAGRAM: asset_name: A plane and two perpendicular lines; asset_slug: plane_perpendicular_bisector; caption: Both crossing lines are perpendicular to AB, but only the one through the midpoint bisects it.; description: A pale parallelogram represents part of a plane containing segment AB. Midpoint M has matching ticks on AM and MB, and a perpendicular line through M has a right-angle square. A second perpendicular line crosses AB away from M, labelled perpendicular only.]

For example, a $14\,\mathrm{cm}$ segment has midpoint $7\,\mathrm{cm}$ from each end. A line through that midpoint meeting the segment at $90^\circ$ is its perpendicular bisector. A sloping line through the midpoint bisects the segment but is not perpendicular to it.

Two lines that look separated in a three-dimensional sketch need not be parallel: they may lie in different planes. Parallel lines must lie in the same plane and never meet when extended.

Apply both conditions to the next description.

Further solids

A solid's name can describe its faces or explain how it was formed. These three solids extend the cube, prism, pyramid, cone and sphere you already know.

A tetrahedron has a triangular base and three triangular side faces, four vertices and six edges. It need not have equilateral faces. A regular tetrahedron is the special case with four equal equilateral faces.

A hemisphere has one flat circular face and one curved surface. The cutting circle has the same radius as the original sphere because the cutting plane passes through its centre. Cutting a sphere away from its centre makes a smaller cap, not a hemisphere.

[DIAGRAM: asset_name: Further solids; asset_slug: tetrahedron_hemisphere_frustum; caption: A tetrahedron has four triangular faces; a hemisphere is half a sphere; a frustum retains parallel bases.; description: Exact projections of a tetrahedron, a hemisphere with its flat circular face, a cone with the removed top shown dashed above its frustum, and a square-pyramid frustum. The two frustums have parallel top and bottom faces of different sizes.]