5.1 Congruency
Definition of Congruency
Two geometric figures are congruent if they have the exact same shape and size. All corresponding sides are equal in length, and all corresponding angles are equal in magnitude.
Triangle Congruency Criteria
- Side-Side-Side (SSS): Three corresponding sides are equal ($S = S = S$).
- Side-Angle-Side (SAS): Two corresponding sides and the enclosed angle are equal ($S = A = S$).
- Angle-Side-Angle (ASA): Two corresponding angles and the enclosed side are equal ($A = S = A$).
- Angle-Angle-Side (AAS): Two corresponding angles and a non-enclosed side are equal ($A = A = S$).
- Side-Side-Angle (SSA): Two corresponding sides and a non-enclosed angle (applicable in specific right-angled scenarios).
- Hypotenuse-Side (RHS): For right-angled triangles, the hypotenuse and one other side are equal.
Isometry and Congruency
Transformations that preserve shape and size are called isometries. Rigid transformations—namely Translation, Reflection, and Rotation—are isometric, meaning the object and its image are always congruent.
5.2 Enlargement
Scale Factor ($k$)
An enlargement is a non-isometric transformation where an object changes size relative to a fixed point called the center of enlargement.
$$\text{Scale Factor, } k = \frac{\text{Distance of Image Point from Center}}{\text{Distance of Object Point from Center}} = \frac{\text{Length of Side of Image}}{\text{Length of Side of Object}}$$
- $k > 1$: Image is larger than the object and lies on the same side of the center.
- $k = 1$: Image is the exact same size as the object.
- $0 < k < 1$: Image is smaller than the object and lies on the same side of the center.
- $-1 < k < 0$: Image is smaller, inverted, and lies on the opposite side of the center.
- $k < -1$: Image is larger, inverted, and lies on the opposite side of the center.
Area of Image under Enlargement
$$\text{Area of Image} = k^2 \times \text{Area of Object}$$
$$\text{Area of Region Covered by Image} - \text{Area of Object} = \text{Area of Shadowed/Shaded Region}$$
5.3 Combined Transformations
Notation and Order of Execution
For a combined transformation represented as $AB$ acting on an object $P$:
$$\text{Object } P \xrightarrow{\quad B \quad} \text{Image } P' \xrightarrow{\quad A \quad} \text{Final Image } P''$$
Transformation $B$ is performed first, followed by transformation $A$.
5.4 Tessellation
A tessellation is a pattern formed by repeating a shape to cover a plane completely without any gaps or overlaps.