Flashcards · 4MA1 · Topic 4

Geometry and trigonometry flashcards

40 cards on Topic 4, subtopics 4.1 to 4.8. Pick a subtopic or study them all.

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Exterior angle of a triangle: It equals the sum of the two interior opposite angles. Example: interior angles 48° and 67° give an exterior angle of 115° at the third vertex.
Alternate and allied angles on parallel lines: Alternate angles are equal; allied (co-interior) angles add to 180°.
Acute, obtuse and reflex angles: Acute: less than 90°. Obtuse: between 90° and 180°. Reflex: between 180° and 360°.
Isosceles triangle: angle fact: The base angles are equal.

Sum of the interior angles of an n-sided polygon: (n − 2) × 180°, the same as (2n − 4) × 90°. A 12-sided polygon: 10 × 180° = 1800°.
Exterior and interior angle of a regular polygon: Exterior = 360° ÷ n. Interior = 180° − exterior. Regular decagon: exterior 36°, interior 144°.
Interior and exterior angles: They add to 180° at each vertex; only the exterior angles always total 360°.
A regular polygon has exterior angle 30°. How many sides, and what is the interior angle?: n = 360° ÷ 30° = 12 sides. Interior angle = 180° − 30° = 150°.

Lines of symmetry and order of rotational symmetry: parallelogram: 0 lines of symmetry; rotational symmetry of order 2.
Lines of symmetry and order of rotational symmetry: rhombus and kite: Rhombus: 2 lines, order 2. Kite: 1 line, order 1.
Symmetry of a regular polygon with n sides: n lines of symmetry and rotational symmetry of order n. A square has 4 and 4.
Congruent shapes: The same shape and the same size. Reflected copies count as congruent.

Bearings: Measured from north, clockwise, with three figures. Back bearing = bearing ± 180°.
Speed, density and pressure formulae: Speed = distance ÷ time. Density = mass ÷ volume. Pressure = force ÷ area (given in the exam).
A car travels 91 km in 1 h 45 min. Find its average speed.: 1 h 45 min = 1.75 h. 91 ÷ 1.75 = 52 km/h
Convert 15 m/s to km/h: Multiply by 3.6 (× 3600 ÷ 1000): 15 × 3.6 = 54 km/h.
Reading scales: Find the value of one small division first. If 5 divisions span 0 g to 100 g, each division is 20 g.

How do you construct the perpendicular bisector of AB?: 1. Set the compasses to more than half of AB. 2. Draw arcs above and below the line from A, then from B, with the same radius. 3. Rule a line through the two crossing points.
How do you construct the bisector of the angle at B?: 1. Draw an arc from B cutting both arms. 2. From those two points draw equal arcs that cross inside the angle. 3. Rule a line from B through the crossing.
Why must you keep construction arcs visible?: The arcs are the evidence of the construction method; rubbing them out removes the marks that depend on them.
Scale drawing: how do you find a real length?: Real length = drawing length × scale factor, then convert units (100 000 cm = 1 km).

Chord, tangent and radius: A chord joins two points on the circle; a tangent touches it once; a radius joins the centre to the circumference.
Circle facts about tangents (both tiers): A tangent is perpendicular to the radius at the point of contact. Tangents from one external point are equal in length.
Perpendicular from the centre to a chord: It bisects the chord (and the line from the centre to the midpoint of a chord is perpendicular to it).
Angle at the centre and angle in a semicircleHigher: The angle at the centre is twice the angle at the circumference on the same arc. The angle in a semicircle is 90°.
Cyclic quadrilateral, and angles in the same segmentHigher: Opposite angles of a cyclic quadrilateral add up to 180°. Angles in the same segment are equal.
Alternate segment theoremHigher: The angle between a tangent and a chord equals the angle in the alternate segment.
Intersecting chords theoremHigher: Chords AB and CD meeting inside the circle at X: AX × XB = CX × XD. Extended to meet outside at P: PA × PB = PC × PD.

How should you write reasons in an angle proof?: Use the full fact, for example "alternate angles are equal" or "corresponding angles are equal", not "Z angles" or "F angles". Give a reason for each stage.
Angles on a straight line, and angles at a point: Angles on a straight line add up to 180°. Angles at a point add up to 360°.
Vertically opposite angles, and angles in a quadrilateral: Vertically opposite angles are equal. Angles in a quadrilateral add up to 360°.
A tangent from P touches a circle of radius 7 cm at T. OP = 25 cm. Find PT.: The tangent is perpendicular to the radius, so OTP has a right angle at T. PT = √(25² − 7²) = √576 = 24 cm.

Pythagoras' theorem, and the hypotenuse with legs 9 and 12: a² + b² = c², where c is the hypotenuse. √(81 + 144) = √225 = 15.
sin, cos and tan in a right-angled triangle: sin = opposite/hypotenuse cos = adjacent/hypotenuse tan = opposite/adjacent These are not on the formulae sheet.
A right-angled triangle has hypotenuse 20 cm and an angle of 35°. Find the side opposite 35°.: opposite = 20 × sin 35° = 11.5 cm (3 s.f.)
The sine rule and the cosine ruleHigher: Sine rule: a/sin A = b/sin B = c/sin C (use with a side and its opposite angle). Cosine rule: a² = b² + c² − 2bc cos A (two sides and the included angle, or three sides).
Area of a triangle with sides 5 cm and 8 cm and included angle 150°Higher: Area = ½ab sin C = ½ × 5 × 8 × sin 150° = 20 × 0.5 = 10 cm²
sin x = 0.6. Find both values of x between 0° and 180°.Higher: x = sin⁻¹ 0.6 = 36.9°, and 180° − 36.9° = 143.1° (1 d.p.), because sin(180° − θ) = sin θ.
Angle of elevation and angle of depressionHigher: Both are measured from the horizontal, up or down. The angle of depression from A to B equals the angle of elevation from B to A.
3D trigonometry: angle between a line and a planeHigher: Use the right-angled triangle made by the line, its projection on the plane and the vertical. Find any missing base length first, often a diagonal, using Pythagoras.