Flashcards · 0580 · Topic 4

Geometry flashcards

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

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4.1 Geometrical terms

Acute, obtuse and reflex angles: Acute: less than 90°. Obtuse: between 90° and 180°. Reflex: between 180° and 360°. (A right angle is exactly 90°.)
Congruent and similar shapes: Congruent: identical in shape and size. Similar: identical in shape only; one is an enlargement (scaled copy) of the other.
Trapezium, kite and rhombus: Trapezium: exactly one pair of parallel sides. Kite: two pairs of equal adjacent sides. Rhombus: four equal sides, opposite sides parallel.
Chord, tangent, sector and segment of a circle: Chord: a straight line joining two points on the circle. Tangent: a straight line touching the circle at one point. Sector: region between two radii and an arc. Segment: region between a chord and an arc.
Number of sides of a pentagon, hexagon, octagon and decagon: Pentagon 5, hexagon 6, octagon 8, decagon 10.
Hemisphere and frustumExtended: Hemisphere: half a sphere. Frustum: a cone or pyramid with the top cut off parallel to the base.
Minor arc and major arcExtended: Minor arc: the shorter part of the circumference between two points. Major arc: the longer part.

4.2 Geometrical constructions

Construct a triangle given its three sides (ruler and compasses only): 1. Draw the longest side with a ruler. 2. Set the compasses to the second length and draw an arc from one end. 3. Set them to the third length and draw an arc from the other end. 4. Join the crossing point to both ends.
Why must you leave the construction arcs on a constructed triangle?: The syllabus says construction arcs must be shown: they are the evidence of the method. Do not rub them out.
Sides 3 cm, 4 cm and 8 cm: why can the triangle not be constructed?: 3 + 4 is less than 8, so the arcs do not cross. If one side is as long as the other two together, or longer, no triangle exists.
Nets: how many faces does the net of a cuboid and of a triangular prism have?: Cuboid: six rectangles. Triangular prism: two triangles and three rectangles.
How do you find a surface area from a net?: Add the areas of all the faces of the net: the surface area is the total area of the net.

4.3 Scale drawings

A map scale is 1 : 25 000. Two places are 4.4 cm apart on the map. What is the real distance?: 4.4 × 25 000 = 110 000 cm = 1100 m = 1.1 km
A lake is 2.4 km long. How long is it on a map of scale 1 : 50 000?: 2.4 km = 240 000 cm; 240 000 ÷ 50 000 = 4.8 cm. Real to map: divide.
Write "1 cm represents 2 km" as a ratio scale: 2 km = 200 000 cm, so 1 : 200 000.
Three-figure bearings: An angle measured clockwise from north, written with three figures. North 000°, east 090°, south 180°, west 270°. 45° is written 045°.
The bearing of B from A: where do you measure from?: Stand at A (the point after "from"), draw a north line at A, and turn clockwise until you face B.
The bearing of Q from P is 063°. Find the bearing of P from Q.: 063° + 180° = 243°. Back bearings differ by 180° (add 180° if under 180°, subtract if 180° or more).

4.4 Similarity

What is true about two similar shapes?: Their corresponding angles are equal and their corresponding sides are in the same ratio (the scale factor).
Two triangles are similar. A side of 4 cm corresponds to 6 cm. Another side of the smaller triangle is 5 cm. Find the matching side.: Scale factor 6 ÷ 4 = 1.5, so 5 × 1.5 = 7.5 cm.
Linear scale factor k: what are the area and volume scale factors?Extended: Area scale factor k². Volume scale factor k³.
AB = 2 cm corresponds to PQ = 5 cm in similar triangles. Triangle ABC has area 12 cm². Find the area of PQR.Extended: k = 2.5, so area factor 2.5² = 6.25. 12 × 6.25 = 75 cm²
Two similar solids have volumes in the ratio 8 : 27. Find the ratio of their lengths and of their surface areas.Extended: Lengths: cube root, 2 : 3. Surface areas: square the length ratio, 4 : 9.
How do you show that two triangles are similar?Extended: Show that two pairs of corresponding angles are equal, giving a geometric reason for each (e.g. alternate angles, a shared angle). The third pair must then be equal too.

4.5 Symmetry

Line symmetry and order of rotational symmetry: Line symmetry: the number of mirror lines. Order of rotational symmetry: how many times the shape fits onto itself in one full turn. Order 1 means no rotational symmetry.
Symmetry of a parallelogram (not a rectangle or rhombus): No lines of symmetry, rotational symmetry of order 2.
Symmetry of a rectangle and of a rhombus (neither a square): Both: 2 lines of symmetry, rotational order 2. Rectangle: the lines join midpoints of opposite sides (not the diagonals). Rhombus: the lines are the diagonals.
Symmetry of a kite (not a rhombus) and of an isosceles trapezium: Both: 1 line of symmetry and rotational order 1.
Symmetry of a regular polygon with n sides: n lines of symmetry and rotational symmetry of order n. A regular hexagon: 6 lines, order 6.
Plane of symmetry of a solidExtended: A plane that cuts the solid into two halves that are mirror images: the 3D version of a line of symmetry.
Planes of symmetry: a cuboid with all edges different, and a cubeExtended: Cuboid (all edges different): 3 planes. Cube: 9 planes.

4.6 Angles

Angles on a straight line, angles at a point, vertically opposite angles: On a straight line: sum to 180°. At a point: sum to 360°. Vertically opposite: equal.
Angles formed with parallel lines: Corresponding angles are equal. Alternate angles are equal. Co-interior angles sum to 180°.
Angle sum of a triangle and of a quadrilateral: Triangle: 180°. Quadrilateral: 360°.
Exterior and interior angle of a regular octagon: Exterior angles sum to 360°, so one exterior angle = 360° ÷ 8 = 45°. Interior angle = 180° − 45° = 135°.
Each interior angle of a regular polygon is 150°. How many sides does it have?: Exterior angle = 180° − 150° = 30°. Number of sides = 360° ÷ 30° = 12.
Sum of the interior angles of a polygon with n sides: (n − 2) × 180°. For a pentagon: 3 × 180° = 540°.
Four exterior angles of an irregular pentagon are 70°, 80°, 60° and 90°. Find the fifth.Extended: Exterior angles of any polygon sum to 360°: 360° − 300° = 60°.

4.7 Circle theorems

The angle in a semicircle: The angle in a semicircle is 90°: a triangle drawn on a diameter, with its third vertex on the circle, has a right angle at that vertex.
The angle between a tangent and a radius: 90°: a tangent meets the radius at right angles at the point of contact.
AB is a diameter and C is on the circle. Angle CAB = 35°. Find angle ABC.: Angle ACB = 90° (angle in a semicircle), so angle ABC = 180° − 90° − 35° = 55°.
Angle at the centre theoremExtended: The angle at the centre is twice the angle at the circumference standing on the same arc.
Angles in the same segmentExtended: Angles in the same segment, standing on the same arc (chord), are equal.
Cyclic quadrilateral theoremExtended: Opposite angles of a cyclic quadrilateral sum to 180°. It applies only when all four vertices lie on the circle.
Alternate segment theoremExtended: The angle between a tangent and a chord equals the angle in the alternate segment.

4.8 Circle theorems II

The perpendicular bisector of a chordExtended: The perpendicular bisector of a chord passes through the centre of the circle.
Equal chordsExtended: Equal chords are equidistant (the same perpendicular distance) from the centre.
Tangents from an external point P touch the circle at A and B. What do you know?Extended: PA = PB (tangents from an external point are equal). Triangle PAB is isosceles, and OAPB is a kite with right angles at A and B.
A chord of length 16 cm is in a circle of radius 10 cm. How far is the chord from the centre?Extended: Use half the chord: √(10² − 8²) = √36 = 6 cm.
Exam trap with chords and PythagorasExtended: Using the whole chord as a side of the right-angled triangle. The triangle uses half the chord and the perpendicular distance from the centre.