Geometry and measures flashcards

Topic 4: Geometry and measures

Angles and parallel lines

Alternate angles are equal. Corresponding angles are equal. Use these names as reasons, not "Z angles" or "F angles".

Angle sums of polygons

Interior angles of an n-sided polygon add to (n − 2) × 180°. Regular polygon: exterior angle = 360° ÷ n; interior angle = 180° − exterior angle.

Conditions for congruent triangles

SSS, SAS (the angle must be between the two sides), ASA, RHS. AAA shows similarity, not congruence.

Arc length and sector area for angle θ and radius r

Arc length = (θ/360) × 2πr Sector area = (θ/360) × πr² Sector perimeter = arc length + 2r. Example: r = 12, θ = 75°: area 30π cm², arc 5π cm.

Pythagoras and the right-angled trig ratios

a² + b² = c², where c is the hypotenuse. sin x = opp/hyp, cos x = adj/hyp, tan x = opp/adj. Use sin⁻¹, cos⁻¹ or tan⁻¹ to find an angle.

Exact trig values for 30°, 45° and 60°

sin 30° = cos 60° = 1/2 sin 60° = cos 30° = √3/2 sin 45° = cos 45° = √2/2 tan 30° = √3/3, tan 45° = 1, tan 60° = √3

Circle theoremsHigher

Angle at the centre = 2 × angle at the circumference. Angle in a semicircle = 90°. Angles in the same segment are equal. Opposite angles of a cyclic quadrilateral add to 180°. Tangent ⊥ radius. Tangents from a point are equal. Perpendicular from the centre bisects a chord. Alternate segment theorem.

Sine rule, cosine rule and area: which to use?Higher

Side and opposite angle known: sine rule, a/sin A = b/sin B. Two sides and the included angle, or three sides: cosine rule, a² = b² + c² − 2bc cos A. Area = ½ab sin C.

Similar shapes with length scale factor k: area and volume factorsHigher

Area scale factor k². Volume scale factor k³. To go back from areas to lengths, square-root; from volumes, cube-root.

How do you prove three points are collinear with vectors?Higher

Show two of the vectors are parallel (one is a multiple of the other) and that they share a point. Parallel alone is not enough.