Flashcards · J560 · Topic 8

Basic geometry flashcards

30 cards on Topic 8, subtopics 8.01 to 8.06. Pick a subtopic or study them all.

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Acute, obtuse and reflex angles: Acute: under 90°. Right: 90°. Obtuse: over 90° but under 180°. Reflex: over 180° but under 360°.
Notation: angle ABC and side a in triangle ABC: In angle ABC the vertex is the middle letter, B. Side a is opposite angle A, so a = BC.
How are parallel and perpendicular lines marked?: Parallel: matching arrows. Perpendicular (meet at 90°): a small square at the corner.
Midpoint of (a, b) and (c, d): ((a + c)/2, (b + d)/2)
Rhombus compared with kite: Rhombus: four equal sides, opposite sides parallel. Kite: two pairs of adjacent equal sides.

How do you construct the perpendicular bisector of AB?: 1. Set compasses wider than half of AB. 2. Draw arcs above and below AB from A, then from B with the same radius. 3. Join the two crossing points. Keep the arcs.
How do you construct the bisector of an angle?: 1. Arc from the vertex cutting both arms at P and Q. 2. Equal arcs from P and from Q crossing at R. 3. Join the vertex to R.
Locus of points equidistant from two points; from two intersecting lines: From two points: the perpendicular bisector of the line joining them. From two lines: the bisector of the angle between them.
Locus of points a fixed distance from a line segment: Two parallel lines joined by two semicircles at the ends.
Shortest distance from a point to a line: The perpendicular distance. Any slanted route is the hypotenuse of a right-angled triangle, so it is longer.
Which angles can you construct with ruler and compasses?: 60° (equilateral triangle), 30° (bisect 60°), 90° (perpendicular) and 45° (bisect 90°).

Angles at a point, on a straight line, and vertically opposite: At a point: sum to 360°. On a straight line: sum to 180°. Vertically opposite angles are equal.
Angles with parallel lines: Alternate angles are equal. Corresponding angles are equal. Co-interior angles sum to 180°.
Exterior angle of a triangle: Equals the sum of the two opposite interior angles. Example: angles 47° and 68° give an exterior angle of 115° at the third vertex.
Alternate angles are 4k + 6 and 6k − 34. Find the angle.: Alternate angles are equal: 4k + 6 = 6k − 34, so 40 = 2k and k = 20. The angle is 86°.

Interior and exterior angles of polygons: Interior angles of an n-sided polygon sum to (n − 2) × 180°. Exterior angles of any polygon sum to 360°. Interior + exterior = 180° at each vertex.
Regular polygon: exterior angle and number of sides: Exterior angle = 360° ÷ n, so n = 360° ÷ exterior angle. Interior angle = 180° − exterior angle.
The interior angles of a polygon sum to 2340°. How many sides?: (n − 2) × 180 = 2340, so n − 2 = 13 and n = 15.
Diagonals of a parallelogram, rhombus and kite: Parallelogram: bisect each other. Rhombus: bisect each other at right angles (and bisect the angles). Kite: perpendicular.
Symmetry of a parallelogram and a rectangle: Parallelogram: no lines of symmetry, rotation symmetry of order 2. Rectangle: 2 lines of symmetry, order 2. Regular n-gon: n lines, order n.

Chord, tangent, arc, sector and segment: Chord: line joining two points on the circumference. Tangent: touches the circle at one point. Arc: part of the circumference. Sector: region between two radii and an arc. Segment: region between a chord and an arc.
Circle theorems: angle at the centre and angle in a semicircleHigher: The angle at the centre is twice the angle at the circumference standing on the same arc. The angle in a semicircle is 90°.
Circle theorems: same segment and cyclic quadrilateralHigher: Angles in the same segment are equal. Opposite angles of a cyclic quadrilateral add to 180°.
Circle theorems: tangent and chordHigher: A radius meets a tangent at 90°. The perpendicular from the centre bisects a chord (and a radius that bisects a chord is perpendicular to it).
Alternate segment theoremHigher: The angle between a tangent and a chord equals the angle in the alternate segment (on the other side of the chord).
A circle has radius 5 cm and a chord of 8 cm. How far is the chord from the centre?Higher: The perpendicular from the centre bisects the chord: half-chord 4 cm. Distance = √(25 − 16) = 3 cm.

Faces, edges and vertices: triangular prism and square-based pyramid: Triangular prism: 5 faces, 9 edges, 6 vertices. Square-based pyramid: 5 faces, 8 edges, 5 vertices.
Prism and pyramid with an n-sided base: faces, edges, vertices: Prism: n + 2 faces, 3n edges, 2n vertices. Pyramid: n + 1 faces, 2n edges, n + 1 vertices.
Check for counting faces, vertices and edges of a flat-faced solid: faces + vertices − edges = 2
Plan compared with elevation: Plan: the view from above. Elevation: the view from the front or side. A standing cylinder has a circle as its plan and a rectangle as its front elevation.