The Theorem of Pythagoras Pythagoras Theorem In a

The Theorem of Pythagoras

Pythagoras’ Theorem In a right angled triangle: c a c 2 = a 2 + b 2 b The longest side, opposite the right angle, is called the hypotenuse

Find x in the diagram below Pythagoras’ Theorem: c 2 = a 2 + b 2 x 15 cm 22 cm Longest side (hypotenuse) = x So x 2 = 152 + 222 = 225 + 484 To find x we find the square root of 709 = 709 So x = 709 = 26 • 63

Worked Example 5 x 9

Worked Example 2 x 6 4

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

Now we’ll see what happens when we need to look for a shorter side Find x 11 cm y 27 cm Pythagoras’ Theorem: c 2 = a 2 + b 2 Longest side (hypotenuse) = 27 (known) When we wanted to find the longest side we added the squares. To find a shorter side we subtract the squares. Longest – ADD Shorter - SUBTRACT x 2 = 272 - 112 = 729 - 121 = 608 x = 608 = 24 • 66

x 11 8

6. 7 3. 5 x

Calculate the value of ‘x’ in each of the examples below x mm 73· 3 mm 12 cm x cm 9· 75 cm 1 2 43 mm 85 mm 7 cm 15· 1 cm 3· 1 m 3 x cm 11· 6 cm 9· 7 cm 1· 62 m xm 4 3· 5 m

Try these examples x 4 x 8 5 x 5 14 3 6. 7 x 18 4. 1 x 5 13 6 x

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

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

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

Pythagoras Problems

A ship leaves port and sails 18 km north. It then turns and sails 15 km west. How far is the ship from the port? The hands of a clock are 8. 4 cm and 5. 8 cm long. How far apart are the tips of the hands at 9 pm?

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

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. 14 m ladder x wall 6 m ground 9 m A wall is 6 m high. A ladder 14 m long rests against the wall with the foot of the ladder 9 m from the bottom of the wall as shown in the diagram. What length of the ladder x hangs over the top of the wall?

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

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 side of length 8 cm. Find the length of the dotted line in the diagram. 6 Find the perimeter of triangle ABC. B 5 3 A 5. 4 9 2. 8 C

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? 70 m The diagram shows an arena 25 m used for animals at the pigs Royal Agricultural Show 42 m at Ingliston. cows Calculate the total length of fencing needed to construct 35 m this arena. goats sheep 16 m

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

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

An aero plane flight BA 2184 leaves an airport and flies 90 miles north and then 80 miles west. At the same time a second aeroplane Ryan 4456 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.

x 12 2 x 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) y By Pythagoras: (7, 9) ● d 2 = 62 + 72 d 2 = 36 + 49 d ● (1, 2) 7 6 d 2 = 85 d = √ 85 x d = 9· 22

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

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

What Is A Converse ? Consider the sentence below: If the angles of the shape add up to 180 o Then the shape is a triangle. To make the converse statement swap around the parts of the statement in the white box: If Then the shape is a triangle. the angles of the shape add up to 180 o 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 360 o Now make the converse statement. If Then the angles add up to 360 o a shape is a square. Can you think of a shape with angles of 360 o 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 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. Theorem Of Pythagoras states: If Then a given triangle is right angled a 2 + b 2 = c 2 for a triangle. Write the converse statement. If Then a 2 + b 2 = c 2 for a triangle. 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 ? 10 m 6 m 8 m 62+82 = 36 + 64 102 100 =100 As 6 2 + 8 2 = 10 2 then by the converse of Pythagoras the triangle is right angled. (1) Which side is the longest side ? 10 m (2) Add the sum of the squares of the two shorter sides. (3) Square the longest side separately. (4) Are the two calculations equal to each other? yes

Is the triangle below right angled ? 11. 5 6. 9 (1) Which side is the longest side ? 11. 5 9. 2 6. 9 2 + 9. 2 2 = 47. 61+84. 64 11. 5 2 132. 25 =132. 25 As 6. 9 2 + 9. 2 2 = 11. 5 2 then by the converse of Pythagoras the triangle is right angled. (2) Add the sum of the squares of the two shorter sides. (3) Square the longest side separately. (4) Are the two calculations equal to each other? yes

Is the triangle below right angled ? 14. 5 8. 1 (1) Which side is the longest side ? 14. 5 10. 8 8. 1 2 + 10. 8 2 = 65. 61+116. 64 14. 5 2 210. 25 =182. 25 As 6. 9 2 + 9. 2 2 11. 5 2 then by the converse of Pythagoras the triangle is not right angled. (2) Add the sum of the squares of the two shorter sides. (3) Square the longest side separately. (4) Are the two calculations equal to each other? No.

Use the converse of Pythagoras to determine if these triangles are right angled or not. (1) 13 5 23. 4 (2) 28. 8 Yes 39 No 12 (3) 41. 5 107. 9 (4) 123. 5 49 Yes 99. 6 117. 6 No