Congruence, similarity, transformations and coordinate geometry

Geometry and measures • Properties and constructions

Flashcards

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How does composition of two reflections in intersecting lines act?

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It produces a rotation about the intersection point through twice the angle between the lines.

Key concepts

What you'll likely be quizzed about

Triangle congruence criteria (SSS, SAS, ASA, RHS)

SSS (side-side-side) declares triangles congruent when all three pairs of corresponding sides are equal. SAS (side-angle-side) declares triangles congruent when two pairs of corresponding sides and the included angle are equal. ASA (angle-side-angle) declares triangles congruent when two pairs of corresponding angles and the included side are equal. RHS (right angle-hypotenuse-side) declares right-angled triangles congruent when the hypotenuse and one other side are equal. The criteria restrict matching parts: SSS and SAS require corresponding sides in the same order; ASA requires the side between the equal angles; RHS applies only to right-angled triangles. Congruence implies equality of all corresponding angles and sides, which permits substitution in geometric arguments and simple proofs.

Similarity and scale factors, including fractional and negative

Similarity requires equal corresponding angles and proportional corresponding sides. A scale factor k relates corresponding lengths: image length = k × original length. A positive k > 1 enlarges, 0 < k < 1 reduces, and k = 1 preserves size. A negative scale factor combines enlargement with a half-turn (rotation by 180°) about the centre of enlargement, reversing orientation. Fractional scale factors produce smaller similar figures with all side lengths multiplied by the fraction. Ratios of areas follow k^2. Similarity enables solving unknown lengths and using proportional reasoning in proofs and coordinate constructions.

Basic transformations and their algebraic effects

Translation shifts every point by the same vector (a, b); coordinates transform by (x, y) → (x + a, y + b). Rotation about the origin by 90° or 180° uses fixed coordinate rules: 90° clockwise (x, y) → (y, -x); 90° anticlockwise (x, y) → (-y, x); 180° (x, y) → (-x, -y). Reflection across axes changes the sign of the relevant coordinate: reflection in x-axis (x, y) → (x, -y); in y-axis (x, y) → (-x, y). Enlargement with centre at the origin multiplies coordinates by k: (x, y) → (kx, ky). An enlargement about a non-origin centre (h, k) transforms via translation to origin, scaling, then translating back. Algebraic rules permit precise constructions on coordinate axes and solving equations that describe images.

Compositions of rotations, reflections and translations

Composing rigid motions produces a single transformation with predictable properties. Two translations combine to one translation by vector addition, so successive translations preserve orientation and distances. Two reflections across parallel lines produce a translation perpendicular to the lines; across intersecting lines produce a rotation about the intersection through twice the angle between the lines. Compositions preserve congruence because rigid motions preserve distances and angles. The order of non-commuting transformations affects the result, so notation and sequence matter when constructing combined maps or deducing invariance.

Coordinate geometry problem solving and constructions

Coordinate geometry converts geometric relations into algebraic equations. The gradient of a line uses rise over run, and perpendicular gradients multiply to -1. Midpoint and distance formulas produce exact coordinates for segment midpoints and lengths. These formulas allow testing congruence (equal distances), similarity (ratios of distances) and verifying right angles (using gradients or Pythagoras). Constructions on the coordinate plane use algebraic transformation rules or vector methods to locate images under rotations, reflections, translations and enlargements. Algebraic methods supply simple proofs by calculation and comparison of coordinates.

Angle facts, Pythagoras and properties of isosceles triangles and quadrilaterals

Angle facts include supplementary, complementary and vertically opposite angles. Pythagoras’ theorem relates right triangle sides: a^2 + b^2 = c^2. In an isosceles triangle, equal sides produce equal base angles; the perpendicular from the apex to the base bisects the base and the apex angle when it is also an altitude. Quadrilateral properties (parallelogram opposite sides equal and parallel; rectangles contain right angles; rhombi have equal sides) combine with triangle results to produce angle and side conjectures. Combining these facts supports short deductive proofs and the derivation of new relationships.

Key notes

Important points to keep in mind

SSS, SAS and ASA require correct matching of corresponding parts; check included angle where needed.

RHS only applies when one angle is exactly 90°.

Similarity preserves angles and scales lengths by k; areas scale by k².

Negative scale factors flip orientation and act like an enlargement plus a 180° rotation.

Translation vectors add component-wise; two translations combine into one.

Two reflections in parallel lines give a translation; in intersecting lines give a rotation.

Use distance and midpoint formulas to test congruence and construct images on coordinate axes.

Use gradients to test parallelism and perpendicularity; perpendicular gradients multiply to −1.

Apply Pythagoras to verify right angles or compute lengths from coordinates.

In an isosceles triangle, equal sides imply equal base angles and the perpendicular from apex bisects the base.