1. Starter Work out the missing lengths for these squares and cuboids
Area = 25cm2
Area = 64cm2
Area = 81 cm2 x x x
Area = 6.25cm2
Vol = 64cm3 x
Vol = 216cm3 y y

2. KS4 Mathematics
S2 Pythagoras’ Theorem

3. Right-angled triangles
A right-angled triangle contains a right angle.
The longest side opposite the right angle is called the hypotenuse.
Review the definition of a right-angled triangle.
Tell pupils that in any triangle the angle opposite the longest side will always be the largest angle and vice-versa.
Ask pupils to explain why no other angle in a right-angled triangle can be larger than or equal to the right angle.
Ask pupils to tell you the sum of the two smaller angles in a right-angled triangle.
Recall that the sum of the angles in a triangle is always equal to 180°. Recall also, that two angles that add up to 90° are called complementary angles. Conclude that the two smaller angles in a right-angled triangle are complementary angles.

4. Identify the hypotenuse
Use this exercise to identify the hypotenuse in right-angled triangles in various orientations.

5. Objective:
By the end of the lesson you should be able to work out the missing side of any right angled triangle using Pythagoras’ Theorem.
Key words:
Hypotenuse
Pythagoras’ theorem
Right angled triangle
Big blue Foundation books
Pages
For the brave pages
PLEASE DRAW THE TRIANGLE AND SHOW YOUR WORKING CLEARLY
ALWAYS THINK ARE YOU TRYING TO FIND A LONGER SIDE OR A SHORTER SIDE

6. The history of Pythagoras’ Theorem
Pythagoras’ Theorem concerns the relationship between the sides of a right- angled triangle.
The theorem is named after the Greek mathematician and philosopher, Pythagoras of Samos.
Explain that the theorem is named after Pythagoras because he, or a member of his society, was the first person known to have formally proven the result.
Although the Theorem is named after Pythagoras, the result was known to many ancient civilizations including the Babylonians, Egyptians and Chinese, at least 1000 years before Pythagoras was born.

7. Pythagoras’ Theorem
Pythagoras’ Theorem states that the square formed on the hypotenuse of a right-angled triangle …
… has the same area as the sum of the areas of the squares formed on the other two sides.
Pythagoras’ Theorem may be expressed as a relationship between areas, as shown here, or a relationship between side lengths.
In fact, the area of any similar shapes may by drawn on the sides of a right-angled triangle. The area of the shape drawn on the hypotenuse, will be equal to the sum of the areas of the shapes drawn on the two shorter sides.
See S9.4 Circles for a problem involving the areas of semi-circles drawn on the sides of a right-angled triangle.

8. Showing Pythagoras’ Theorem
Drag the vertices of the triangle to change the lengths of the sides and rotate the right-angled triangle.
Ask a volunteer to come to the board and use the pen tool to demonstrate how to find the area of each square. For tilted squares this can be done by using the grid to divide the squares into triangles and squares. Alternatively, a larger square can be drawn around the tilted square and the areas of the four surrounding triangles subtracted.
Reveal the areas of the squares and verify that the area of the largest square is always equal to the sum of the areas of the squares on the shorter sides.

9. Pythagoras’ Theorem c2 a2 b2
If we label the length of the sides of a right-angled triangle a, b and c as follows,
then the area of the largest square is c × c or c2. c2
The areas of the smaller squares are a2 and b2. c a2 a
We can write Pythagoras’ Theorem as
This slide shows how Pythagoras’ Theorem can be written as a relationship between the side lengths of a triangle with sides a, b and c, where c is the hypotenuse.
Ask pupils to tell you what a2 is equal to. (c2 – b2).
Ask pupils to tell you what b2 is equal to. (c2 – a2). b b2
c2 = a2 + b2

10. S2.4 Finding unknown lengths
Contents
S2 Pythagoras’ Theorem A
S2.1 Introducing Pythagoras’ Theorem A
S2.2 Identifying right-angled triangles A
S2.3 Pythagorean triples A
S2.4 Finding unknown lengths A
S2.5 Applying Pythagoras’ Theorem in 2-D A
S2.6 Applying Pythagoras’ Theorem in 3-D

11. Pythagoras’ Theorem
Pythagoras’ Theorem states that for a right-angled triangle with a hypotenuse of length c and the shorter sides of lengths a and b c a
c2 = a2 + b2 b
We can use Pythagoras’ Theorem
to check whether a triangle is right-angled given the lengths of all the sides,
Stress that Pythagoras’ Theorem is only true for right-angled triangles.
If we are given the lengths of all three sides of a triangle, therefore, we can use Pythagoras’ Theorem to check whether of not the triangle is a right-angled triangle by squaring the length of the two shorter sides, adding the squares together and checking whether or not this is equal to the hypotenuse squared. This is demonstrated in S2.2 Identifying right-angles.
If we are given the lengths of two of the sides in a right-angled triangle, we can also use Pythagoras’ Theorem to find the lengths of the unknown sides.
to find the length of a missing side in a right-angled triangle given the lengths of the other two sides.

12. Finding the length of the hypotenuse
Use Pythagoras’ Theorem to calculate the length of side a. a
5 cm
12 cm
Using Pythagoras’ Theorem,
a2 =
a2 =
Talk through the calculation to find the length of the hypotenuse.
Stress that if a2 = 169, to find a we need to find the square root of 169.
If pupils do not know the square root of 169, ensure that they are able to locate and use the  key on their calculators. Encourage pupils to learn at least the first 15 square numbers.
Pupils should also be encouraged to look at the triangle to make sure that their answer seems about right compared to the other two given sides.
a2 = 169
a = 169
a = 13 cm

13. Finding the length of the hypotenuse
Use Pythagoras’ Theorem to calculate the length of side PR.
0.7 m Q
2.4 m R
Using Pythagoras’ Theorem
PR2 = PQ2 + QR2
Substituting the values we have been given,
PR2 =
This example uses the letters at each vertex to label the sides.
Stress to pupils that they must start by stating Pythagoras’ Theorem for the given triangle and show every step in their calculation.
They should also draw a diagram if one has not been given.
Again encourage pupils to think about whether the answer seems sensible given the lengths of the other two sides.
PR2 =
PR2 = 6.25
PR = 6.25
PR = 2.5 m

14. Finding the length of the shorter sides
Use Pythagoras’ Theorem to calculate the length of side a.
26 cm
10 cm a
Using Pythagoras’ Theorem,
a = 262
a2 = 262 – 102
a2 = 676 – 100
Explain that when we use Pythagoras’ Theorem to find the length of one of the shorter sides, we have to subtract the square of the given shorter side from the square of the hypotenuse.
a2 = 576
a = 576
a = 24 cm

15. Finding the length of the shorter sidesA
Use Pythagoras’ Theorem to calculate the length of side AC to 2 decimal places.
5 cm B C
8 cm
Using Pythagoras’ Theorem
AB2 + AC2 = BC2
Substituting the values we have been given,
52 + AC2 = 82
AC2 = 82 – 52
PR2 = 39
PR = 39
PR = 6.24 cm

16. Finding the length of the shorter sides
Use Pythagoras’ Theorem to calculate the length of side x to 2 decimal places.
15 cm
7 cm x
Using Pythagoras’ Theorem,
x = 152
x2 = 152 – 72
x2 = 225 – 49
x2 = 176
x = 176
x = cm

17. S2.5 Applying Pythagoras’ Theorem in 2-D
Contents
S2 Pythagoras’ Theorem A
S2.1 Introducing Pythagoras’ Theorem A
S2.2 Identifying right-angled triangles A
S2.3 Pythagorean triples A
S2.4 Finding unknown lengths A
S2.5 Applying Pythagoras’ Theorem in 2-D A
S2.6 Applying Pythagoras’ Theorem in 3-D

18. Finding the lengths of diagonals
Pythagoras’ Theorem has many applications. For example, we can use it to find the length of the diagonal of a rectangle given the lengths of the sides.
Use Pythagoras’ Theorem to calculate the length of the diagonal, d.
10.2 cm
13.6 cm d
d2 =
d2 =
d2 = 289
d = 289
d = 17 cm

19. Finding the lengths of diagonals
Use Pythagoras’ Theorem to calculate the length d of the diagonal in a square of side length 7 cm. d
7 cm
Using Pythagoras’ Theorem
d2 =
d2 =
d2 = 98
d = 98
d = 9.90 cm (to 2 d.p.)

20. Finding the height of an isosceles triangle
Use Pythagoras’ Theorem to calculate the height h of this isosceles triangle.
5.8 cm h
8 cm
Using Pythagoras’ Theorem in half of the isosceles triangle, we have
h = 5.82
h2 = 5.82 – 42
5.8 cm
4 cm h
Explain that the height divides the isosceles triangle into two equal parts and so the length of the base in the right-angled triangle must be half of the length of the base of the original isosceles triangle.
Ask pupils when it might be important to know the perpendicular height of a triangle, for example, when trying to work out its area.
h2 = – 16
h2 = 17.64
h = 17.64
h = 4.2 cm

21. Finding the height of an equilateral triangle
Use Pythagoras’ Theorem to calculate the height h of an equilateral triangle with side length 4 cm.
4 cm h
Using Pythagoras’ Theorem in half of the equilateral triangle, we have
h = 42 h
4 cm
2 cm
h2 = 42 – 22
Explain that like the isosceles triangle, the height divides the equilateral triangle into two equal parts and so the length of the base in the right-angled triangle must be half of the length of the base of the original equilateral triangle.
h2 = 16 – 4
h2 = 12
h = 12
h = 3.46 cm (to 2 d.p.)

22. Ladder problem
Change the length and the position of the ladder to generate various problems.

23. Flight path problem
Use Pythagoras’ Theorem to calculate the distance of the aeroplane from the starting point.

24. Contents S2 Pythagoras’ Theorem A S2.1 Introducing Pythagoras’ Theorem
S2.2 Identifying right-angled triangles A
S2.3 Pythagorean triples A
S2.4 Finding unknown lengths A
S2.5 Applying Pythagoras’ Theorem in 2-D A
S2.6 Applying Pythagoras’ Theorem in 3-D

25. Pythagorean triples
A triangle has sides of length 3 cm, 4 cm and 5 cm.
Does this triangle have a right angle?
Using Pythagoras’ Theorem, if the sum of the squares on the two shorter sides is equal to the square on the longest side, the triangle has a right angle. =
= 25
Explain that a Pythagorean triple is three whole numbers that obey Pythagoras’ Theorem. Many Pythagorean triples where known to people in ancient times.
= 52
Yes, the triangle has a right-angle.
The numbers 3, 4 and 5 form a Pythagorean triple.

26. Pythagorean triples
Three whole numbers a, b and c, where c is the largest, form a Pythagorean triple if,
a2 + b2 = c2
3, 4, 5 is the simplest Pythagorean triple.
Write down every square number from 12 = 1 to 202 = 400.
Use these numbers to find as many Pythagorean triples as you can.
Write down any patterns that you notice.
Pupils should be able to find six Pythagorean triples using these numbers. These are shown on the next slide.
They may notice that some of these are multiples of the 3, 4, 5 triangle.

27. Pythagorean triples How many of these did you find? 9 + 16 = 25
= 52
3, 4, 5
= 100
= 102
6, 8, 10
= 169
= 132
5, 12, 13
= 225
= 152
9,12, 15
= 289
= 172
8, 15, 17
Pupils should notice that the Pythagorean triples that have not been circled are multiples of the the 3, 4, 5 Pythagorean triple.
Tell pupils that a primitive Pythagorean triple is a Pythagorean triple that is not a multiple of another Pythagorean triple.
Ask pupils to explain why any multiple of a Pythagorean triple must be another Pythagorean triple (using similar triangles).
= 400
= 202
12, 16, 20
The Pythagorean triples 3, 4, 5;
5, 12, 13
and 8, 15 17
are called primitive Pythagorean triples because they are not multiples of another Pythagorean triple.

28. Similar right-angled triangles
The following right-angled triangles are similar. 10 ? 15 6 9 8 12 ?
Remind pupils that the corresponding lengths in two mathematically similar shapes are always in the same ratio. That is, one shape is an enlargement of the other.
Ask pupils if they can work out which Pythagorean triple both these triangles are based on.
For the first triangle, 6 and 8 share a common factor of 2. Dividing these lengths by 2, gives us 3 and 4.
For the second triangle, 9 and 15 share a common factor of 3. Dividing these lengths by 3, gives us 3 and 5.
These triangles are both enlargements of a right-angled triangle with sides of length 3, 4 and 5.
Use this to find the lengths of the missing sides.
Alternatively, we could use a scale factor of 3/2 to scale from the smaller triangle to the larger triangle, or a scale factor of 2/3 to scale from the larger triangle to the smaller triangle.
Find the lengths of the missing sides.
Check that Pythagoras’ Theorem holds for both triangles.

29. Similar right-angled triangles
The following right-angled triangles are similar. 10 ? 15 6 9 8 12 ?
Check the lengths of the missing sides by showing that Pythagoras’ Theorem holds for both triangles. = =
= 100
= 225
= 102
= 152