Year 9 Mathematics notes part 3:

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1 Year 9 Mathematis notes part 3: Pythagoras 3.1 (p.109) Angles 3.1 (p.106) Constrution 1.3 (p.23) Loi 6.3 (p.280) Contents Pythagoras theorem... 1 Proving Pythagoras theorem... 3 Angle rules... 4 Constrution methods... 7 Drawing a triangle, given 2 sides and an angle Drawing a triangle given the lengths of 3 sides Biseting an angle... 9 Drawing the perpendiular bisetor of a line segment... 9 Names of shapes Pythagoras theorem The lengths of the sides of a right-angled triangle are related by Pythagoras s theorem: a b where is the length of the longest side. b To find one of the shorter sides, take side: a b 2 b from eah a Examples 1

2 (1) A builder is building a roof. The roof is 6 m wide and 1.4m high at the apex. How long should the rafters be? m 3 m 3 m m (do [ ] ANS = on alulator) (2) A joiner is making a door frame for a door. The door is 500mm wide and 40mm thik. The aperture needs to be slightly wider than this so the door does not jam when opening. How wide does it need to be? 500 mm 40 mm mm, he needs to make the frame at least 1.6mm wider than the door. (3) I buy a sheet of plasterboard 1.2 m wide and want to get it home in my ar. The ar is only 1 m wide so it will have to sit at an angle. How high will one side be? a 1.2 m 1 m a a m 2

3 Aside: there are many right angled triangles that have sides with integer (whole number) lengths. The simplest of these has side in the ratio 3:4:5 (so 3m, 4m and 5m or 6m, 8m and 10m). These sets of three numbers that satisfy triplets. a b are known as Pythagorean Other ommon triplets are (5, 12, 13), (7, 24, 25) and (8, 15, 17). Proving Pythagoras theorem There are dozens of ways of proving it. This is the easiest to remember. Draw a small square inside a larger one. a b b a a b b a The triangles are all ongruent beause they all have the same hypotenuse () and the same angles. If I all the short triangle sides a and b, the big square has sides of length a+b Area of big square = (area of small square) + (area of the 4 triangles) a b 4 ab 2 Expand the brakets: a 2ab b 2ab Now take 2ab from eah side: 3

4 a b Angle rules Angles on a straight line add to x x x A C B A B C Angle ABC is the aute or obtuse (not reflex) angle between lines AB and BC. 1. Angles around a point add to 360. Whenever possible show your working by writing an equation and solving it. x x = 360 x x When straight lines ross, the vertially opposite angles are equal: x x 3. When a straight line uts a pair of parallel lines: (a) all the aute angles are equal (b) all the obtuse angles are equal 4

5 Names for pairs of angles: Corresponding (F) angles are equal Alternate (Z) angles are equal Supplementary angles add to 180 Exterior angles The exterior angles of any polygon add to 360, regardless of the number of sides. A regular polygon has n sides of equal length and rotation symmetry of order n about its entre point. The exterior angles are all = 360. n e.g. A regular polygon has an exterior angle = 72. How many sides does it have? , n 360 n 5 sides, it is a pentagon. 72 5

6 Interior angles Angles inside a triangle add to 180 The sum of all the interior angles inside any polygon with n sides is 180n 2 sine one an draw n-2 triangles inside it, e.g. for a hexagon (n = 6) one an draw n-2 = 6-2 = 4 triangles and they add up to = 720. If it is a regular polygon, eah interior angle will be 180 n 2. n (One an more easily get the same result using exterior angle). Degrees, minutes, seonds. 1 degree = 60 minutes of ar ( ) = 3600 seonds of ar ( ), so 1 45 =

7 Constrution methods What information do we need before we an draw a triangle? Drawing a triangle, given 2 sides and an angle. Usually one side is already drawn on the answer sheet (if not, draw it). Your length must be aurate to the nearest millimetre. Use the protrator to measure the angle. Make a little mark, then (using the ruler) draw a long faint line through it and the starting point. Set your ompasses to draw a irle with the length of the third side. Draw an ar that uts your line. Put a line between the orner points. Example 3 Draw the following triangle aurately 5 m 30 6 m Draw 2 lines at 30 to eah other. Measure and mark a point 6 m along one, 5 m along the other. Then draw lines between the points. 7

8 Example 4 Draw the following triangle aurately 51 mm mm As before, start by drawing two lines at 30. Measure a 6 m bottom edge and mark its ends. Set the ompasses to draw a irle of 4 m radius and draw a irular ar entered on point B This will ross line AC at two points. Draw a line through whihever looks most like the example in the sketh in the question. 51 mm Drawing a triangle given the lengths of 3 sides. Draw one side Use ompasses to draw an ar with exatly the right radius for eah other side. Where the ars ross is the third orner. Example 5 Draw a triangle with sides of length 7, 6 and 5 m. 6 m radius 5 m radius 8

9 Biseting an angle Using the ompasses, draw an ar entered on the point where the lines ross. Your ar rosses eah line one. Put the ompass spike on one rossing point and draw an ar (an be same radius as first, but not essential). Now very arefully without hanging the ompass width, put the spike on the other rossing point and draw another ar (must be same radius as the one before). Draw a straight line through the intersetions. Drawing the perpendiular bisetor of a line segment A line segment is a line between two points. A perpendiular bisetor goes through the mid-point of this line and uts it at right angles. You are expeted to onstrut it using ompasses. The examiners will look for your onstrution lines. Both ars must be exatly the same radius. This gives you the lous of points equidistant from A and B. (A lous is a line defined by a series of points). 9

10 Names of shapes Two-dimensional shapes Two-dimensional shapes often have names ending in -gon. The name identifies the number of sides. One side: Cirle Ellipse 3 sides (triangle): Equilateral = all sides same length, all angles 60. Isoseles = two sides same length, two angles equal Salene = 3 sides all different lengths Right-angle = one angle is 90 Types of quadrilateral (4 sides): retangle rhombus (a sheared square) square parallelogram (a sheared retangle) kite trapezium (two parallel sides) arrowhead quadrilateral (anything else) 10

11 Shapes with more sides Pentagon (5 sides) Hexagon (6 sides) Heptagon (7 sides) Otagon (8 sides) Also onsider: For a quadrilateral If the diagonals biset eah other it must be a parallelogram (speial ases retangle, rhombus, square) If the diagonals are perpendiular it must be a kite (speial ase = rhombus, square) Shapes will tessalate if one an make their internal angles add to

12 Loi There are three ways of defining a line: use an equation for a line on a graph (e.g y = x-1 ) define a starting point and a diretion from London, drive Northwards at A- level you will see how we use vetors to define lines use a rule for the distane from other points or lines. A lous is a line defined as a set of points that follow some rule. (i) Constant distane from a point e.g. 1 m from A. This defines a irle: A 1 m Similarly less than 1 m from A is the region inside the irle, more than 1 m is the region outside the irle. (ii) onstant distane from a line segment A line segment is a short line. The lous runs beside the line, then makes a semiirle around the ends: lous A line segment AB B (iii) equidistant from points A and B The lous will be the perpendiular bisetor of an imaginary line segment AB: lous A B 12

13 (iv) An ellipse is a lous that we get if we have a loop of string passing around two points A, B and our penil (point C). The sum of these distanes AC+BC is onstant. The points A and B are eah a fous (plural foi). If the ellipse were a shiny surfae, light oming from A would be foussed to make an image at B. Just to help you remember: the Earth s orbit is an ellipse. The Sun is at one fous. When losest, we are 91 million miles from the Sun; when furthest away, 93 million miles. If you do Further Mathematis A-level, you will find out about other loi, the parabola and hyperbola. Extension: in Engineering, the shape of gear teeth is very important. This shape is alled an involute: it is the lous of the end of a string as it unrolls from a irle. It lets the gears rotate at onstant speed, i.e. with a onstant ratio between the rpm (revolutions per minute) for the left and right-hand gears. See this animation. 13