Grade 11 Spatial Geometry Review

Flashcards covering spatial geometry concepts including plane intersections, line-plane intersections, collinearity, segment ratio formulas, Thales's theorem, tangent planes, perpendicularity, and cross-sections.

Intersection line of two planes

The straight line formed where two non-parallel planes intersect, determined by finding 2 common points between the planes.

Intersection point of a line and a plane

A single point $A$ where line $d$ meets plane $(P)$, satisfying $A \in d$ and $A \in (P)$, written as $d \cap (P) = A$.

Collinear points

Three or more points that all lie on the same straight line $d$.

Segment division ratio formula

If point $M$ divides segment $AB$ such that $AM : MB = m : n$, then $\frac{AM}{AB} = \frac{m}{m+n}$ and $\frac{MB}{AB} = \frac{n}{m+n}$.

Thales's Theorem in Spatial Geometry

In $\triangle ABC$, if line $MN$ is parallel to side $BC$ with $M \in AB$ and $N \in AC$, then $\frac{AM}{AB} = \frac{AN}{AC} = \frac{MN}{BC}$.

Converse of Thales's Theorem

In $\triangle ABC$, if points $M \in AB$ and $N \in AC$ satisfy $\frac{AM}{AB} = \frac{AN}{AC}$, then $MN \parallel BC$.

Tangent plane

A plane $(P)$ that touches a sphere $(S)$ at exactly 1 point $A$, known as the point of tangency.

Distance from center to tangent plane

The perpendicular distance from the center $O$ of a sphere to its tangent plane $(P)$, which equals the radius $R$ ($d(O, (P)) = OA = R$).

Perpendicularity condition for line and plane

A line $d$ is perpendicular to plane $(P)$ if $d$ is perpendicular to two intersecting lines $a$ and $b$ contained in $(P)$, written as $d \perp (P)$.

Cross-section (Thiết diện)

The closed 2D planar polygon figure formed by the intersection of a cutting plane with a 3D solid figure.

Parallel cross-section property

If a cutting plane $(P)$ is parallel to base $(ABCD)$ of a solid, the resulting cross-section is similar to the base shape.

Cross-section ratio property

For a pyramid cross-section with line $MN$ parallel to base side $AB$, the side ratios satisfy $\frac{SM}{SA} = \frac{SN}{SB} = \frac{MN}{AB}$.

Procedure for finding a cross-section

A 4-step method: 1) Identify the cutting plane, 2) Find intersection points with solid edges, 3) Connect points lying on the same face, 4) Close the figure.

Two-plane intersection rule

Two non-parallel planes intersect along a single line defined by 2 common points.

Line-plane intersection rule

A line and a plane intersect at a single common point.