Pythagoras’s Theorem One of the most important rules you will meet in mathematics – and it’s 2500 years old….. x 2 + y 2 = z 2.

Pythagoras’s Pythagoras’s TheoremTheorem

One of the most important One of the most important rules you will meet in rules you will meet in mathematics – and it’s 2500 mathematics – and it’s 2500 years old…..years old…..

x2 + y2 = z2

Aims for today’s Aims for today’s lesson:lesson: Briefly revisit some angle problems Briefly revisit some angle problems

and the idea of similar trianglesand the idea of similar triangles Understand a special connection Understand a special connection

between the lengths of the sides of between the lengths of the sides of some triangles;some triangles;

Know that this connection is called Know that this connection is called Pythagoras’s TheoremPythagoras’s Theorem

Use the Theorem in some Use the Theorem in some problems..problems..

Quick recap QuizQuick recap Quiz

1.1. What is the total (sum) of all What is the total (sum) of all the exterior angles of ANY the exterior angles of ANY polygon?polygon?

2.2. What name do we give to two What name do we give to two angles between two parallel angles between two parallel lines which form a ‘Z’ shape?lines which form a ‘Z’ shape?

3.3. What is the sum of all the What is the sum of all the angles inside a quadrilateral?angles inside a quadrilateral?


4.4. The diagram shows part of a The diagram shows part of a regular polygon that has 8 regular polygon that has 8 sides (an octagon). What are sides (an octagon). What are the sizes of the angles x and the sizes of the angles x and y?y?

xy


5.5. Look at the diagrams below. Look at the diagrams below. Find the sizes of angles a, b, Find the sizes of angles a, b, c and d.c and d.

b a

2873

c

d

6. What is the name given to angles 73 and c in the diagram above?

1

2 3

4A

BC

D

EF

5

67

89

1011

12

7.Give the three-letter code for angle 3.

Quick recap Quiz - Quick recap Quiz - ANSWERSANSWERS

1.1. What is the total (sum) of all the What is the total (sum) of all the exterior angles of ANY polygon? exterior angles of ANY polygon? (360)(360)

2.2. What name do we give to two What name do we give to two angles between two parallel angles between two parallel lines which form a ‘Z’ shape? lines which form a ‘Z’ shape? (Alternating)(Alternating)

3.3. What is the sum of all the angles What is the sum of all the angles inside a quadrilateral? inside a quadrilateral? (360)(360)


4.4. The diagram shows part of a The diagram shows part of a regular polygon that has 8 regular polygon that has 8 sides (an octagon). What are sides (an octagon). What are the sizes of the angles x and the sizes of the angles x and y?y?

xy

X = 360 ÷ 8 = 45° and y = 180 – 45 = 135°

Quick recap Quiz - Quick recap Quiz - ANSWERSANSWERS

5.5. Look at the diagrams below. Look at the diagrams below. Find the sizes of angles a, b, Find the sizes of angles a, b, c and d.c and d.

b a

2873

c

d

6. What is the name given to angles 73 and c in the diagram above?

a = b = 76° c=73

d=107

F-shape, so CORRESPONDING

1

2 3

4A

BC

D

EF

5

67

89

1011

12

6.Give the three-letter code for angle 3.

Angle 3 = CFD or DFC

SIMILAR SHAPESSIMILAR SHAPES – – a remindera reminder Shapes are similar when one is an Shapes are similar when one is an

ENLARGEMENT of the otherENLARGEMENT of the other The enlargement must be achieved by The enlargement must be achieved by

multiplying all the sides by the same multiplying all the sides by the same amount, called the amount, called the scale factorscale factor

Two shapes are NOT similar if we just Two shapes are NOT similar if we just add the same amount onto all the add the same amount onto all the sides.sides.

If the two shapes are identical size, If the two shapes are identical size, they are called CONGRUENT instead.they are called CONGRUENT instead.

Examples:Examples:1. 1. Below are two SIMILAR triangles. Below are two SIMILAR triangles.

Work out the length of the sides Work out the length of the sides marked x and y, and the angle a.marked x and y, and the angle a.

4cm

4.5cm 18cm

12cm

52°

y

x

a

Examples:Examples:1. 1. Below are two SIMILAR triangles. Below are two SIMILAR triangles.

Work out the length of the sides Work out the length of the sides marked x and y, and the angle a.marked x and y, and the angle a.

4cm

4.5cm 18cm

12cm

52°

y

x

a

The 4cm is enlarged to 12cm, so the scale factor is 3. So x = 4.5 x 3 = 13.5 cm and y is 18 ÷ 3 = 6 cm. Angles never change, so a = 52°

KEY QUESTION: Quite a hard GCSE-type problem:

Timpkins Builders make wooden frames for roofs on new houses.

In the diagram of the wooden frame shown below, PQ is parallel to BC.

QP

CB600 cm

200 cm

400 cm

N O T TO SC ALE

A

Calculate length PQ using similar triangles.

Your task:Your task: For the following triangles, use a compass For the following triangles, use a compass

and a ruler to draw them accurately, and a ruler to draw them accurately, putting the middle sized length as the putting the middle sized length as the base;base;

Then, Then, for each trianglefor each triangle, multiply each side , multiply each side length by itself (square it), writing the length by itself (square it), writing the three answers you get inside the triangle three answers you get inside the triangle you have drawn.you have drawn.

Do you notice anything about four of the Do you notice anything about four of the triangles and the values you work out? triangles and the values you work out?

TRIANGLE 1:TRIANGLE 1: 3cm, 4cm, 5cm3cm, 4cm, 5cmTRIANGLE 2TRIANGLE 2: 5cm, 12cm, 13cm: 5cm, 12cm, 13cmTRIANGLE 3: 10cm, 8cm, 6cmTRIANGLE 3: 10cm, 8cm, 6cmTRIANGLE 4: TRIANGLE 4: 3.5cm, 12cm, 12.5cm3.5cm, 12cm, 12.5cmTRIANGLE 5:TRIANGLE 5: 4cm, 7cm, 9.5cm4cm, 7cm, 9.5cm

WHICH OF THESE TRIANGLES IS THE ODD ONE OUT

?

What you What you shouldshould have have found:found: For For anyany right-angled triangleright-angled triangle the the

longest sidelongest side (called the (called the HYPOTENUSEHYPOTENUSE) squared ) squared equals the equals the totaltotal of the other two sides of the other two sides squared!!squared!!

This rule is calledThis rule is called PYTHAGORAS’S PYTHAGORAS’S THEOREMTHEOREM

It won’t work if the triangle has not It won’t work if the triangle has not got a right angle…..Like number 5!got a right angle…..Like number 5!

Using the method:Using the method:

EXAMPLEEXAMPLE: Work out the length : Work out the length marked x in this triangle:marked x in this triangle:

x cm9 cm

40 cm

Using the method:Using the method:EXAMPLE 2EXAMPLE 2: Work out the length : Work out the length

marked x in this triangle (give marked x in this triangle (give your answer to 1 d.p)your answer to 1 d.p)

x cm7.3 cm

12.8 cm

Here’s myHere’s my specialspecial method:method:Step 1: Write the three sides in order of size – like Step 1: Write the three sides in order of size – like

this:this:

7.37.3 12.812.8 x x (miss out (miss out cm)cm)

Step 2: put ‘squares’ onto each number – like this:Step 2: put ‘squares’ onto each number – like this:

7.37.322 12.812.822 xx22

Step 3: put a + and an = in the two gaps – like Step 3: put a + and an = in the two gaps – like this:this:

7.37.322 + + 12.812.822 = = xx22x cm7.3 cm

12.8 cm

HYPOTENUSE

(make sure it’s at the end)

Here’s myHere’s my specialspecial method:method:Step 4: Work out the two parts you can – like this:Step 4: Work out the two parts you can – like this:

53.29 + 163.8453.29 + 163.84 = = xx22

Step 5: Now add the first two answers – like this:Step 5: Now add the first two answers – like this:

217.13217.13 = = xx22

Step 6: Now we need to know what number Step 6: Now we need to know what number squared actually gives 217.13. For this we need squared actually gives 217.13. For this we need the square root key – it looks like this: the square root key – it looks like this: √√

x = √217.13x = √217.13

x = 14.7 cmx = 14.7 cm to to 1dp1dp

x cm7.3 cm

12.8 cm

…….and finally .and finally (but (but crucial!):crucial!):Always Always CHECKCHECK your answer looks right. It your answer looks right. It

has has gotgot to be bigger (longer) than the to be bigger (longer) than the other two sides….other two sides….

WHY????WHY????Because it’s supposed to be the Because it’s supposed to be the

HYPOTENUSE -which is the longest side!!HYPOTENUSE -which is the longest side!!

So x = 14.7 cm is probably So x = 14.7 cm is probably OK.OK. x cm7.3 cm

12.8 cm

Now Now YOUYOU try this one: try this one:Question 3:Question 3: Work out the length marked x in Work out the length marked x in

this triangle (give your answer to 1 d.p)this triangle (give your answer to 1 d.p)

x cm6.6 cm

11.9 cm

Answer:Answer:6.66.6 11.911.9 xx

6.66.622 11.911.922 xx22

6.66.622 + + 11.911.922 = = xx22

43.56 + 141.6143.56 + 141.61 = = xx22

185.17185.17 = = xx22

x = √185.17x = √185.17

x = 13.6 cmx = 13.6 cm to 1dp to 1dp

CHECK: Does it look right?: Does it look right?x cm6.6 cm

11.9 cm

Now a selection for Now a selection for YOU:YOU:Questions:Questions: Work out the length marked x in Work out the length marked x in

these triangles (give your answer to 1 d.p)these triangles (give your answer to 1 d.p)

x cm6.6 cm

4.5 cm

Q1 3.7 cm

15 cm

x cm

Q2

Q3

d cm

7.4 cm

6.1 cm

ANSWERS:ANSWERS:

Q1: Q1: x = √63.81 = x = √63.81 = 8.0 cm8.0 cm (to 1 dp) (to 1 dp)

Q2:Q2: x = √238.69 = x = √238.69 = 15.4 cm15.4 cm (to 1 (to 1 dp)dp)

Q3:Q3: d = √91.97 = d = √91.97 = 9.6 cm9.6 cm (to 1 dp) (to 1 dp)

BUTBUT…what if x is …what if x is NOTNOT the the longest side??longest side??EXAMPLE EXAMPLE Work out the length marked x in Work out the length marked x in


x cm

17.3 cm

13.6 cm

Well, we stick with the same method as before!!

So now for the So now for the specialspecial method:method:Step 1: Write the three sides in order of size – Step 1: Write the three sides in order of size –

like this:like this:

xx 13.613.6 17.3 17.3

Step 2: put ‘squares’ onto each number – like Step 2: put ‘squares’ onto each number – like this:this:

xx22 13.613.622 17.317.322

Step 3: put a + and an = in the two gaps – like Step 3: put a + and an = in the two gaps – like this:this:

xx22 + + 13.613.622 = = 17.317.32217.3 cm

x cm

13.6 cm

HYPOTENUSE

(Again, it’s at the end)

Here’s myHere’s my specialspecial method:method:Step 4: Work out the two parts you can – like this:Step 4: Work out the two parts you can – like this:

xx22 + 184.96+ 184.96 = = 299.29299.29Step 5: Now Step 5: Now subtractsubtract these two answers – like this: these two answers – like this:

xx22 = 299.29 – 184.96 = 299.29 – 184.96 xx22 = 114.33 = 114.33

Step 6: Now we need to know what number squared Step 6: Now we need to know what number squared actually gives 114.33. For this we need the actually gives 114.33. For this we need the square root key – it looks like this: square root key – it looks like this: √√

x = √114.33x = √114.33x = 10.7 cmx = 10.7 cm to 1dp to 1dp17.3 cmx cm

13.6 cm

…….and finally .and finally (but (but crucial!):crucial!):Now again Now again CHECKCHECK your answer looks your answer looks

right. It has right. It has gotgot to be smaller than the to be smaller than the hypotenuse…hypotenuse…

WHY????WHY????Because the HYPOTENUSE is the longest Because the HYPOTENUSE is the longest

side!!side!!

So x = 10.7 cm is probably So x = 10.7 cm is probably OK.OK.

x cm

17.3 cm

13.6 cm

Now Now YOUYOU try this one: try this one:Question 3:Question 3: Work out the length marked x in Work out the length marked x in


x cm12.5 cm

8.9 cm

Answer:Answer:xx 8.98.9 12.512.5

xx22 8.98.922 12.512.522

xx22 + + 8.98.922 = = 12.512.522

xx22 + 79.21 + 79.21 = = 156.25156.25

xx22 = = 156.25 – 79.21156.25 – 79.21

xx22 == 77.0477.04

x = √77.04x = √77.04

x = 8.8 cmx = 8.8 cm to 1dp to 1dp

CHECK: Does it look right?: Does it look right?

x cm

12.5 cm

8.9 cm

Now a selection for Now a selection for YOU:YOU:Questions:Questions: Work out the length marked x in Work out the length marked x in

these triangles (give your answer to 1 d.p)these triangles (give your answer to 1 d.p)

x cm

9.6 cm 14.5 cm

Q1 4.1 cm

13.7cm

x cmQ2

Q3

t cm

17 cm

24 cm

ANSWERS:ANSWERS:

Q1: Q1: x = √118.09 = x = √118.09 = 10.9 cm10.9 cm (to 1 (to 1 dp)dp)

Q2:Q2: x = √170.88 = x = √170.88 = 13.1 cm13.1 cm (to 1 (to 1 dp)dp)

Q3:Q3: d = √287 = d = √287 = 16.9 cm16.9 cm (to 1 dp) (to 1 dp)

Now to recap…….Now to recap……. Look at the problem on the next slideLook at the problem on the next slide

It’s like what you could get at It’s like what you could get at Foundation levelFoundation level…….…….

And shows how you might be asked And shows how you might be asked to apply to apply Pythagoras’s theoremPythagoras’s theorem

A A GCSE-type GCSE-type question:question:A boat leaves a harbour and sails A boat leaves a harbour and sails due due

North for 18KmNorth for 18Km, then turns , then turns East and sails East and sails for a distance of 25kmfor a distance of 25km. How far is the . How far is the direct route back to the Harbour?direct route back to the Harbour?

x km

25km

18km

SolutionSolution::

x km

25km

18km 18 25 = x

182 252 = x2

182 + 252 = x2

324 + 625 = x2

949 = x2

x = 30.8km

Does the answer LOOK right??

Thank-youThank-you to to Pythagoras!!Pythagoras!!

Pythagoras’s Theorem One of the most important rules you will meet in mathematics – and it’s 2500 years old….. x 2 + y 2 = z 2.

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