(You will have covered Pythagoras Theorem at an earlier stage.) Calculating the length of the HYPOTENUSE of a right angled triangle

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1 ythagoras hapter 4 evision of ythagoras Theorem (You will have covered ythagoras Theorem at an earlier stage.) ythagoras was a famous Greek Mathematician who discovered an amazing connection between the three sides of any right angled triangle. He discovered that the 3 sides of a right angled triangle were connected mathematically by the formula :- c b c2 = a 2 + b2 a alculating the length of the HYOTENUSE of a right angled triangle Example :- alculate the length of the hypotenuse of this right angled triangle. Solution :- egin with :- x 2 = x 2 = x 2 = 185 x = 11 cm 185 = alculating the length of one of the SMLLE sides in a..t. Example :- alculate the length of the side marked a in this right angled triangle. *note Solution :- egin with :- 1 a 2 = a 2 = a 2 = 243 a = 9 cm 243 = a cm Exercise alculate the lengths of the missing sides in the following right angled triangles :15 cm (a) (b) 2. (a) alculate the height of the triangle. 26 cm (b) Now calculate its area. y cm 20 cm 19 cm 3. (c) Shown is an isosceles triangle. alculate the area of this rectangle :- (d) 7 6 m 4 1 mm 3 9 mm 75 cm zm 9 8 m w mm N5 - hapter 4 72 cm this is page 26 ythagoras

2 4. alculate the perimeter of this right angled triangle :- 9. Shown is an isosceles triangular prism. 6 5 cm 15 cm 4 cm 5 cm x m 5. alculate the value of x, which indicates the length of the sloping 14 5 m side of this trapezium. 9 7 m 6 4 m (a) alculate the height of the triangle, represented by the dotted line. (b) alculate the volume of the prism. 10. When a boy was asked to calculate the value of x, he proceeded as follows : 6. This shape consists of a rectangle with an isosceles triangle attached to its end. x 2 = x 2 = x 2 = cm x = 149 = 12 2 cm 7 cm 24 cm 20 cm Explain in words, when the boy looked at his answer and at the triangle, why he should have known that his answer had to be wrong. L cm (a) alculate the total length (L) of the figure. (b) Now calculate its area. 11. Shown is a trapezium with a line of symmetry. 36 mm 96 mm 7. Shown are the points ( 3, 4) and G(6, 3). y x 42 mm alculate the perimeter of the above trapezium. 12. fortune teller has a magic glass globe. It rests embedded in a wooden plinth as shown. The plinth measures 32 cm by 7 cm. The diameter of the globe is 30 cm. Draw a coordinate diagram, plot the two points and calculate the length of the line. 8. Draw a new set of axes, plot the 2 points ( 1, 8) and (6, 4) and calculate the length of the line. 7 cm 30 cm h cm 24 cm alculate the overal height, h of the figure. N5 - hapter 4 this is page 27 ythagoras

3 The ONESE of ythagoras Theorem ythagoras Theorem only works on a right angled triangle. We can use ythagoras Theorem in reverse to actually prove that a triangle is right angled. Example :- Look at triangle opposite We can prove it is right angled as follows :- 5 2 cm 6 5 cm 3 9 cm Write down the 3 sides :- = 5 2, = 3 9, = 6 5. Square each side :- 2 = 27 04, 2 = 15 21, 2 = dd the two smaller squares together : = = heck if this is the same value as the largest square : = = 2. We say that, by the ONESE of ythagoras Theorem, the triangle is proven to be right angled at. Exercise heck if this triangle is right angled at. opy and complete : cm 7 5 cm 3. Decide which of these are or are not right angled triangles :- (a) 84 mm 91 mm (b) 35 mm 9 6 m 20 4 m 18 7 m 2 = 18 2 = 324, 2 = =... 2 =... 2 = groundsman wishes to make sure the football pitch is rectangular = =... = 2 by the onverse of ythagoras Theorem, triangle must be r... a... at 2. W Show that this triangle 6 6 cm is NOT right angled. 8 U 11 1 cm i.e. (Show that UW 2 + W 2 U 2 ) 84 m To check, he measures the diagonal length. Is the pitch rectangular? 5. Has this flagpole been erected correctly, so that it is vertical? 105 m 13 5 m 63 m 10 8 m N5 - hapter 4 this is page 28 ythagoras 8 1 m

4 ythagoras Theorem applied to 3-Dimensional roblems ythagoras Theorem only works on a right angled triangle, but right angled triangles appear in 3 Dimensional situations as well. S Example :- alculate the length of the space diagonal of this cuboid. Note :- face diagonal is one joining the opposite vertices of any rectangular face of the cuboid (e.g. D). space diagonal is one joining one vertex of the cuboid to the furthest away vertex (e.g. ). 5 cm D 6 cm Solution :- The answer is found by following 2 steps :- Step 1 :- Space diagonal is the longest side in..t. D. To find, we must first find D in..t. D. D 2 = 2 + D 2 D 2 = = = 100 D = 100 = Step 2 :- Now find, the longest side in..t. D. 2 = D 2 + D 2 2 = = = 125 D = 125 = 11 2 cm Exercise 4 3 U X 1. (a) alculate the length of the face diagonal EG of this cuboid. (b) Now calculate the length of the space diagonal EX. H G W 3 cm 12 cm E 5 cm F 75 cm 2. F G alculate the length of the face diagonal of this water tank, and then calculate the length of the space diagonal H. 60 cm E 80 cm H 3. Make a sketch of this shoe box and show, using two steps, how to calculate the length of its space diagonal. (You may want to letter the vertices). 15 cm Y 1 30 cm 4. W X alculate the length of the space diagonal Y of this cube. U T S N5 - hapter 4 this is page 29 ythagoras

5 5 Shown is a square based pyramid D. Height M = 9 cm. 8. cone has a base diameter of and its sloping edges are 13 cm long. (a) alculate the height of the cone. 13 cm D (b) Now calculate the volume of the cone. M 6 cm 9. This empty tin of McTivies biscuits measures 24 cm by 24 cm by 16 cm high. (a) alculate the length of the diagonal. (b) Write down the length of M. (c) alculate the length of the sloping edge. 6. This opcorn box is in the shape of a pyramid. Would this wooden rod, 40 cm long, be able to fit in the box and still allow the tin to be fully closed with its lid on? 13 cm olume prism = 1 3 rea base x height 10. Just as you can plot a point in 2-dimensions using 2 coordinates (4, 3), you can plot points in 3-dimensions like (4, 3, 1). (See below). z The square top has sides, and the sloping edge is 13 cm long. S (a) alculate the length of the diagonal of the open top. (b) alculate the height of the box. D y 5 (c) alculate the volume of the box, in cm The side face of this wedge is in the shape of a right angled triangle. O 4 3 x 6 cm 12 cm (a) alculate the length of the face diagonal of the base of the wedge. (b) Now calculate the length of the diagonal of the sloping edge. (The red line). In the above figure, is given by (4, 3, 1). (4 right, 3 back and 1 up). is parallel to the x-axis. The cuboid measures 8 by 3 by 5 boxes. (a) Write down the coordinates of the other 7 points making up the cuboid. (b) alculate the length of the diagonal. (c) alculate the length of space diagonals. (d) Harder. alculate the length of the line OS. N5 - hapter 4 this is page 30 ythagoras

6 emember emember...? Topic in a Nutshell 1. alculate the lengths of the sides marked x (a) (b) and y, correct to 3 significant figures. 7 1 cm 11 6 m 13 8 m y m 7 1 cm 2. lot the points ( 5, 7) and (6, 1) on a coordinate diagram and calculate the length of the line. 3. Use the onverse of ythagoras Theorem to decide which, if any, of these triangles is/are right angled :- (a) (b) (c) Y 24 m 12 4 cm 96 mm 40 mm 32 m 9 3 cm Z 39 m 104 mm X 15 5 cm 4. This figure looks like a kite, but is it really one? rove whether it is or is not a kite. 39 mm 95 mm 39 mm 52 mm 65 mm 5. girl made a simple model house out of a cardboard box. (a) alculate the length of the red dotted line. 20 mm (b) alculate the length of the space diagonal of the box. 30 cm 16 cm 6. Shown is a right angled triangular prism. (a wedge). Use ythagoras Theorem twice to calculate the length of the sloping dotted blue line. 17 cm 11 cm 7. Shown is a square based pyramid and a cone. y calculating the height of both, decide which has the greater volume and by how much. 15 cm 12 cm N5 - hapter 4 this is page 31 ythagoras

7 hapter 4 nswers hapter 4 - ectors Exercise a 11 3 cm b 11 7 cm c 6 19 m d 5 66 mm 2. a 24 cm b 240 cm cm cm 5. x = 8 6. a 4 b 960 cm boxes boxes 9. a 6 cm b 60 cm3 10. x, being smaller side, should end up less than mm cm Exercise = = = 2 y the onverse of ythagoras Theorem, it IS a T = = = Therefore, it IS NOT a T 3. a yes b no = = = 1052 y the onverse of ythagoras Theorem, it IS a T = = = y the onverse of ythagoras Theorem, it IS a T and the flagpole could be vertical. emember, emember 1. a 10 0 cm b 7 48 m boxes 3. a = = =1042 y the onverse of ythagoras Theorem, it is a T b = = = 392 Therefore, it IS NOT a T c = = = y the onverse of ythagoras Theorem, it is a T = = 4225 = 652 This means the angle between the diagonals = 90 ecause the 2 smaller parts of one diagonal are equal, this means the longer diagonal is a line of symmetry, which means it IS a kite. 5. a 34 cm b 39 4 cm Height of pyramid = 13 9 cm, Height of cone = olume of pyramid = cm3, olume of cone = cm3 one has bigger volume by 5 cm3 approximately Exercise a 13 cm b 13 3 cm 2. = 100 cm H = 125 cm cm cm 5. a 8 49 cm b 4 24 cm c 9 95 cm 6. a 14 1 cm b 10 9 cm c 364 cm 3 7. a 14 4 cm b 15 6 cm 8. a 12 cm b 314 cm3 9. Space diagonal = 37 5 cm. 40 cm rod is too long to fit in 10. a (12, 3, 1), 12, 6, 1), D(4, 6, 1), (4, 6, 6) (4, 3, 6), (12, 3, 6), S(12, 6, 6) b 8 54 units c 9 90 units d 14 7 N5 - hapter 4 this is page 32 ythagoras