KS3 Maths · Year 8 · Chapter 6: Pythagoras' theorem · Lesson 2

Using Pythagoras' theorem to solve problems

Year 8About 70 minutesCalculator allowed45 marks

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Quick questions to start.

A1
Needed today
Work out $1 7^{2} - 8^{2}$ .
[1 mark]
Answer
A2
Needed today
Write $90$ to $1$ decimal place.
[1 mark]
Answer
A3
Last lesson
A right-angled triangle has shorter sides $9$ cm and $40$ cm. How long is the hypotenuse?
[1 mark]
Answer
A4
Year 7
Change $2.4$ km into metres.
[1 mark]
Answer

Example, then your turn

Year 8

Your teacher works through each example with you. Then try the one beside it. First find the right angle, then decide: is the side you want the hypotenuse, or a shorter side?

Example 1

A ship sails $9$ km due west, then $7$ km due north. How far is it now from where it started? Give your answer to $1$ decimal place.

Hint The two legs meet at a right angle. The direct distance is the hypotenuse.

B1
A plane flies $140$ km due south, then $95$ km due east. How far is it from where it started? Give your answer to $1$ decimal place.
[2 marks]
Not drawn accurately
Answer $\dots$ km

Example 2

A $5.4$ m ladder leans against a wall. Its foot is $1.6$ m from the wall. How far up the wall does it reach? Give your answer to $1$ decimal place.

Hint The ladder is the hypotenuse. The height is a shorter side, so subtract.

B2
A $7.2$ m ladder reaches $6.9$ m up a wall. How far is its foot from the wall? Give your answer to $1$ decimal place.
[2 marks]
Not drawn accurately
Answer $y = \dots$ m

Example 3

Ana flies a kite on a $40$ m string. The kite is directly above a point $18$ m from Ana. She holds the string $1.2$ m above the ground. How high is the kite above the ground? Give your answer to $1$ decimal place.

Hint The right-angled triangle starts at Ana's hand, not at the ground. Find its height, then add

1.2

B3
A balloon is tied to the top of a $1.4$ m post by a $30$ m rope. The balloon is directly above a point $13$ m from the post. How high is the balloon above the ground? Give your answer to $1$ decimal place.
[3 marks]
Not drawn accurately
Answer $h = \dots$ m

Example 4

$A B$ is a chord of a circle, centre $O$ . $A B = 10$ cm and $O M = 4$ cm, where $M$ is the mid-point of $A B$ . Work out the radius and the diameter, to $1$ decimal place.

Hint

O M

meets

A B

at a right angle, and

A M

is half of

A B

. The radius

O A

is the hypotenuse.

B4
A circle, centre $O$ , has radius $9$ cm. A chord $A B$ is $14$ cm long. How far is the chord from $O$ ? Give your answer to $1$ decimal place.
[3 marks]
Not drawn accurately
Answer $x = \dots$ cm

Practice

Year 8

C1
Two sides of a right-angled triangle are $9$ cm and $14$ cm. Work out the two possible lengths of the third side. Give your answers to $1$ decimal place.
[3 marks]
C2
In triangle $X Y Z$ , angle $Y = 9 0^{\circ}$ , $X Z = 7.25$ m and $X Y = 4.18$ m. Work out the length of $Y Z$ . Give your answer to $2$ decimal places.
[2 marks]
Answer $Y Z = \dots$ m

C3
(a) Show that $12$ , $35$ , $37$ is a Pythagorean triple. (b) Write down a Pythagorean triple whose largest number is $74$ .
[3 marks]
C4
A ramp rises $45$ cm over a level distance of $3.2$ m. Work out the length, $x$ , of the ramp in metres. Give your answer to $2$ decimal places.
[3 marks]
Not drawn accurately
Answer $x = \dots$ m

Exam-style questions

GCSE Foundation

D1
A $5$ m ladder reaches $4.8$ m up a wall. Tom says, "If I move the foot of the ladder $0.5$ m closer to the wall, the top will go $0.5$ m higher."
Is Tom right? You must show your working.
[3 marks]
Not drawn accurately

D2
$A B C D$ is a rectangular field. Kai walks from $A$ to $C$ along the edges, through $B$ . Mia walks straight across the field from $A$ to $C$ . How much further does Kai walk than Mia? Give your answer to $1$ decimal place.
[4 marks]
Not drawn accurately
Answer $\dots$ m
D3
A boat sails from a harbour $H$ $8$ km due north to a buoy $B$ . It then sails $11$ km due east to an island $I$ . It then sails straight back to $H$ .
Not drawn accurately
(a)
Work out the distance from $I$ to $H$ . Give your answer to $1$ decimal place.
[2 marks]
Answer $\dots$ km
(b)
The boat sails at $12$ km/h for the whole trip. How long does the trip take? Give your answer in hours and minutes, to the nearest minute.
[3 marks]
Answer $\dots$ hours $\dots$ minutes
Total for D3: 5 marks
D4
A circle has its centre at $C (- 4, 3)$ . It passes through the point $P (5, - 2)$ .
(a)
Work out the radius of the circle. Give your answer to $1$ decimal place.
[2 marks]
Answer $\dots$ units
(b)
Does the circle pass through the point $Q (1, 9)$ ? You must show your working.
[2 marks]
Total for D4: 4 marks

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
Two boats leave a harbour at the same time. One sails due north at $10$ km/h. The other sails due east at $24$ km/h. How long is it until they are $39$ km apart?
[4 marks]
Hint 1 · What to try
After $1$ hour, how far has each boat sailed? How far apart are they then?
Hint 2 · The first line
After $1$ hour they are $10$ km and $24$ km from the harbour, so $1 0^{2} + 2 4^{2} = 26$ km apart. Every hour they get $26$ km further apart.
Hint 3 · The full method
$39 \div 26 = 1.5$ hours, which is $1$ hour $30$ minutes.
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can draw the right-angled triangle in a problem.
I can decide whether I need the hypotenuse or a shorter side.
I can solve problems with ladders, journeys, kites and chords of circles.
I can give an answer to a sensible degree of accuracy.

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