Chapter 7 : Pythagoras Theorem (Part 2)

At the end of Part 2 of this chapter, you will learn about:

The applications of Pythagoras Theorem to right angled triangles

Prior Knowledge:

Pythagoras theorem and Pythagorean triple

Real-Life applications of Pythagoras Theorem

Pythagoras’ theorem helps to calculate the length of the diagonal

connecting two given straight lines. This application of the theorem

is generally used in architecture, carpentry and other construction

works.

Recall:

Pythagoras Theorem

Applications of Pythagoras Theorem

Pythagoras Theorem can be applied in real life. For example, we have the case of a

ladder leaning against a wall and forming a right-angled triangle as shown in the

picture below.

In any right-angled triangle, the square

of the length of the hypotenuse is equal to

the sum of the squares of the lengths of

the other two sides.

𝒄𝟐 = 𝒂𝟐 + 𝒃𝟐

Also, carpenters and masons use Pythagoras theorem in their construction works to

ensure that their buildings or constructions stand straight by using right-angled

triangles.

Example 1:

Find the length of the diagonal of a rectangle with length 24 cm and width 10 cm.

Solution:

Using Pythagoras Theorem,

24 𝑐𝑚

10 𝑐𝑚 𝑎

𝑎2 = 242 + 102

𝑎2 = 576 + 100 = 676

𝑎 = √676 = 26 𝑐𝑚

The length of the diagonal is 26 cm.

Example 2:

A ladder leans against a wall. If the ladder reaches 6 m up the wall and its foot is 8

m from the base of the wall, find the length of the ladder.

Solution:

Let the length of the ladder be 𝑥 cm.

Using Pythagoras Theorem,

𝑥2 = 82 + 62

𝑥2 = 64 + 36 = 100

Diagram not drawn to scale

𝑥 = √100 = 10 𝑚

The length of the ladder is 10 m.

Example 3:

The mast of a yacht is tied to a 13 m cable attached to its top. The cable is 5 m

from the base of the mast on the deck of the yacht. Find the height of the mast.

Solution:

Let the height of the mast be 𝑥 cm.

Using Pythagoras Theorem,

132 = 𝑥2 + 52

169 = 𝑥2 + 25

𝑥2 = 169 − 25 = 144

𝑥 = √144 = 12 𝑚

The height of the mast is 12 m.

13 𝑐𝑚

𝑥 𝑐𝑚

5 𝑐𝑚

Example 4:

A 25-cm drinking straw fits exactly into

a can as shown in the figure. The

diameter of the can is 7 cm. find the

height of the can.

Solution:

Let ℎ be the height of the can.

Using Pythagoras Theorem,

252 = ℎ2 + 72

625 = ℎ2 + 49

ℎ2 = 625 − 49 = 576

ℎ = √576 = 24 𝑚

The height of the can is 24 cm.

Exercises to practise: Exercise 7D

1. A rectangle has dimensions 20 cm by 15 cm. Find the length of the diagonal.

2. A rectangle has length 7 cm and the length of the diagonal is 10 cm. Find the

width of the rectangle.

3. Find the length of the diagonal of a square of side 6 cm.

25 𝑐𝑚

ℎ 𝑐𝑚

7 𝑐𝑚

4. The diagonal of a square is 10 cm. Find the length of each side of the square,

leaving your answer in √ .

5. A man walks 1000 m up the side of a hill. If he has climbed a vertical distance

of 600 m, find the horizontal distance through which he has travelled.

6. The dimensions of a swimming pool are 40 metres and 30 metres. Calculate

the length between the opposite corners.

7. One diagonal of a rhombus is 30 cm. Find the length of the other diagonal if

each side of the rhombus is 17 cm.

8. A ladder leans against a wall. If the ladder is 26 m long and its foot rests 24

m from the foot of the wall, find how far up the wall the ladder reaches.

9. A cycling track is in the shape of a right-angled triangle such that the lengths

of its two shorter sides are 10 m and 24 m. Find the perimeter of the track.

10. The diagonals of a rhombus are of lengths 12 cm and 16 cm. Find the length

of its sides.

11. In the diagram, XYZ is an isosceles triangle. XY = XZ = 10 cm, YZ = 12 cm

and YM = MZ.

Calculate

(a) The length of YM

(b) The length of XM

M Z Y

X

12 cm

(c) The area of triangle XYZ

(d) The perimeter of triangle XYZ.

12. In the triangle ABC, AB = AC = 26 m. D is the midpoint of BC. Find the

length of BC and hence find the area of triangle ABC.

13. An airplane flies 7 km south and then 24 km east. Find how far it is from its

starting point.

14. PQR is an equilateral triangle of side 2 cm. Find the perpendicular distance

from P to QR.

15. A wire of length 13 m is attached to the top of a flag post, which stands 5 m

above ground level as shown in the diiagram.

Find the value of 𝑥.

A

10 m

D

B C

5 m

13 m

𝑥

16. A man runs diagonally across from one corner of a rectangular field 400 m

by 300 m to the opposite corner in a straight line at a speed of 10 metres per

second. Find the time taken for him to complete this distance.

17. State whether triangle PQR is a right-angled traingle or not. Justify your

answer.

In case it is a right-angled triangle, state which angle is the right angle,

justifying your answer.

18. PQRS is a rectangle in which PQ = 25 cm and PS = 7 cm. T is a point on PQ

such that RST is an isosceles triangle whose equal sides are RT and ST. Find

the length of RT.

19. The trapezium in the diagram is right-angled at Q and S and PQ is parallel to

SR.

If PQ = 30 cm, PS = 12 cm and QR =37 cm, find

(a) RS

(b) The area of the trapezium

(c) The perimeter of the trapezium

40 m

41 m

9 m

R

Q

P

S

Q

P

R

37 cm 12 cm

30 cm

20. The diagram shows a circle centre O, radius 8 cm. OAB is a right-angled

triangle. Calculate the length of CB.

Summary:

𝒄𝟐 = 𝒂𝟐 + 𝒃𝟐

In any right-angled triangle,

the square of the length of

the hypotenuse is equal to

the sum of the squares of the

lengths of the other two

sides.

C

B

O

A

8 cm

15 cm

Links for practice:

http://www.bevs.k12.oh.us/Downloads/WS%20-

%20Pythagorean%20Theorem-blizzard%20bag1.pdf

http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Pythag/pythag.html

http://www.friesian.com/pythag.html

Answers to Exercise 7D

1. Length of diagonal = 25 cm

2. Width of rectangle = √51 𝑐𝑚

3. Length of diagonal = √72 𝑐𝑚

4. Length of square = √50 𝑐𝑚

5. Horizontal distance = 800 m

6. Length between opposite corners = 50 metres

7. Length of other diagonal = 16 cm

8. The ladder reaches 10 m up the wall.

9. Perimeter of track = 60 m

10. Length of rhombus = 10 cm

11. (a) YM = 6 cm

(b) XM = 8 cm

(c) Area of triangle XYZ = 48 cm2

(d) Perimeter of triangle XYZ = 32 cm

12. BC = 48 m and Area of triangle ABC = 240 cm2

13. 25 km

14. √3 𝑐𝑚

15.

16. 50 seconds

17. Yes triangle PQR is a right-angled triangle as 9, 40 and 41 form the

Pythagorean triple.

PR is the longest side, so angle Q is the right angle.

18. RT = 24 cm

19. (a) RS = 65 cm

(b) Area of trapezium = 570 cm2

(c) Perimeter of trapezium = 144 cm

20. CM = 9 cm

𝑥 = 12 𝑚