The Theorem of Pythagoras

The longest side, opposite the right angle, is called the hypotenuse Pythagoras’ Theorem In a right angled triangle: c2 = a2 + b2 a b c The longest side, opposite the right angle, is called the hypotenuse

Find x in the diagram below Pythagoras’ Theorem: c2 = a2 + b2 15 cm 22 cm x Longest side (hypotenuse) = x So x2 = 152 + 222 = 225 + 484 = 709 To find x we find the square root of 709 So x = 709 = 26•63

Worked Example 5 x 9

Worked Example 2 4 x 6

1 2 3 4 Pythagoras’ Theorem: c2 = a2 + b2 Find x in each diagram below 12 cm x 10·5 cm 2 x 3 cm 1 8 cm x 9·1 m 15·4 m 3 x 72 mm 43 mm 4

Pythagoras’ Theorem: c2 = a2 + b2 Now we’ll see what happens when we need to look for a shorter side Find x 11 cm 27 cm y Pythagoras’ Theorem: c2 = a2 + b2 Longest side (hypotenuse) = 27 (known) x2 = 272 - 112 When we wanted to find the longest side we added the squares. = 729 - 121 = 608 To find a shorter side we subtract the squares. x = 608 Longest – ADD Shorter - SUBTRACT = 24•66

8 x 11

6.7 x 3.5

2 1 3 4 Calculate the value of ‘x’ in each of the examples below x mm 12 cm x cm 2 1 43 mm 9·75 cm 85 mm 7 cm 15·1 cm 3·1 m 9·7 cm 1·62 m 3 4 x m 3·5 m 11·6 cm x cm

Try these examples x 8 4 x 5 5 3 x x 14 18 13 x 6 x 6.7 4.1

Now we’ll look at two worded problems which involve the use of Pythagoras

( 4•2 – 3) This house extension has a roof as shown. Calculate the width of the extension. 3 m 4•2 m 4 m Width extract the right-angled triangle write in the lengths of each side 4 ( 4•2 – 3) 1•2 w w2 = 42 - 1•22 = 16 - 1•44 = 14•56 x = 14•56 = 3•82

A flagpole is supported by 2 ropes as shown in the diagram A flagpole is supported by 2 ropes as shown in the diagram. Find the height of the flagpole. 25 m 14 m 1 m extract the right-angled triangle write in the lengths of each side h 25 m 7 m half of 14 m h2 = 252 - 72 = 625 - 49 = 576 h = 576 The flagpole = 24 + 1 = 25 m = 24

Pythagoras Problems

A ship leaves port and sails 18km north. It then turns and sails 15km west. How far is the ship from the port? The hands of a clock are 8.4cm and 5.8cm long. How far apart are the tips of the hands at 9pm?

Calculate the value of x in the diagram shown. Give your answer correct to 1 decimal place. 6cm x 11cm 16cm 19cm The diagram shows a rhombus with diagonals 19cm and 9cm long. Calculate the length of the side of the rhombus. 9cm

x On square paper plot the points A(-4,2), B(7,-3) and C(3,6) and form a triangle. Use Pythagoras to find the perimeter of the triangle. x 14m wall ladder 6m ground 9m A wall is 6m high. A ladder 14m long rests against the wall with the foot of the ladder 9m from the bottom of the wall as shown in the diagram. What length of the ladder x hangs over the top of the wall?

Use Pythagoras to find the length of the line marked 4 10 x Calculate the values of x and y in the diagram shown. 8 y Use Pythagoras to find the length of the line marked with an x. 14 5 9 x

If A(1,2), B(5,1) and C(4,-3) are 3 vertices of a rectangle ABCD, find the coordinates of D. Draw the rectangle on square paper. Use Pythagoras to find the lengths of the sides and the diagonals of the rectangle. An aeroplane leaves an airport and flies 98 km north east. It then turns and flies 114 km south east. How far is the aeroplane from the airport?

An equilateral triangle has Find the length of the dotted side of length 8cm. Find the length of the dotted line in the diagram. 6 B 5 3 Find the perimeter of triangle ABC. A 5.4 9 C 2.8

The size of a television screen is given as the size of the diagonal length of the screen. If a television screen has a length of 32 inches and a breadth of 27 inches, what size screen does the television have? 70m The diagram shows an arena used for animals at the Royal Agricultural Show at Ingliston. Calculate the total length of fencing needed to construct this arena. 25m goats pigs 16m 42m cows sheep 35m

The lengths of the diagonals of a kite are 16cm and 6cm respectively. Calculate the lengths of the sides of the kite 10cm 6 cm 16cm Which of the points A(4,5), B(3,-5) and C(-6,2) is furthest away from the origin?

Use Pythagoras to find x. 6cm 4cm Use Pythagoras to find x. 3cm 12cm Calculate the length of the side marked x in the diagram shown. Give your answer correct to 1 decimal place. 6cm x 5cm 12cm

An aero plane flight BA2184 leaves an airport and flies 90 miles north and then 80 miles west. At the same time a second aeroplane Ryan4456 leaves the airport and flies at the same speed, 60 miles south and then 110 miles east. How far apart are the two planes at this time? A circle has a centre C(5,2). The point A(8,6) lies on the circumference of the circle. Calculate the radius of the circle. Which of the points P(9,0), Q(2,6) and R(1,1) lie on the circle.

12 x 2x The diagram shows a right angled triangle. Calculate the lengths of the sides of the triangle.

Find the distance between the points (1,2) and (7,9) x y By Pythagoras: (7,9) d 2 = 62 + 72 ● d 2 = 36 + 49 d 7 d 2 = 85 d = √85 ● 6 (1,2) d = 9·22

Find the distance between the points (-5,-2) and (6,7) x y By Pythagoras: d 2 = 92 + 112 d 2 = 81 + 121 ● (6,7) d 2 = 202 d d = √202 9 d = 14·2 ● 11 (-5,-2)

The Converse Of Pythagoras. 14m 19m 23m Is the triangle right angled or not ? ©Microsoft Word clipart

What Is A Converse ? Consider the sentence below: If the angles of the shape add up to 180o Then the shape is a triangle. To make the converse statement swap around the parts of the statement in the white box: If the shape is a triangle. Then the angles of the shape add up to 180o This is the converse statement.

Not all converse statements are true. Consider the sentence below: If a shape is a square Then the angles add up to 360o Now make the converse statement. If the angles add up to 360o Then a shape is a square. Can you think of a shape with angles of 360o which is not a square ? Any closed quadrilateral.

What Goes In The Box ? Write the converse statement and decide if the converse of the statement is true or false. (1) If a shape is a square then the shape has parallel sides. False (2) If a number is even then the number divides by two exactly. True (3) If you have thrown a double six with a dice then your score with the dice is twelve. True (4) If you have thrown a three and a four then your total score is seven with a dice. False

The Converse Of Pythagoras. The Theorem Of Pythagoras states: If a given triangle is right angled Then a2 + b2 = c2 for a triangle. Write the converse statement. If a2 + b2 = c2 for a triangle. Then a given triangle is right angled This converse is true and allows us to find right angled triangles.

Testing For A Right Angled Triangle. Is the triangle below right angled ? (1) Which side is the longest side ? 6m 8m 10m 10m (2) Add the sum of the squares of the two shorter sides. 6 2 + 8 2 102 (3) Square the longest side separately. = 36 + 64 100 =100 (4) Are the two calculations equal to each other? As 6 2 + 8 2 = 10 2 then by the converse of Pythagoras the triangle is right angled. yes

Is the triangle below right angled ? 6.9 9.2 11.5 (1) Which side is the longest side ? 11.5 (2) Add the sum of the squares of the two shorter sides. 6.9 2 + 9.2 2 11.5 2 (3) Square the longest side separately. = 47.61+84.64 132.25 =132.25 (4) Are the two calculations equal to each other? As 6.9 2 + 9.2 2 = 11.5 2 then by the converse of Pythagoras the triangle is right angled. yes

Is the triangle below right angled ? 8.1 10.8 14.5 (1) Which side is the longest side ? 14.5 (2) Add the sum of the squares of the two shorter sides. 8.1 2 + 10.8 2 14.5 2 (3) Square the longest side separately. = 65.61+116.64 210.25 =182.25 (4) Are the two calculations equal to each other? As 6.9 2 + 9.2 2  11.5 2 then by the converse of Pythagoras the triangle is not right angled. No.

Use the converse of Pythagoras to determine if these triangles are right angled or not. (2) 39 28.8 23.4 (1) 13 5 12 Yes No (3) 107.9 41.5 99.6 (4) 49 117.6 123.5 Yes No