Solve Problems Using Pythagoras' Theorem

In this worksheet, students will apply Pythagoras' theorem to solve simple problems.

Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   AQA, Eduqas, Pearson Edexcel, OCR

Curriculum topic:   Geometry and Measures, Mensuration

Curriculum subtopic:   Mensuration and Calculation, Triangle Mensuration

Difficulty level:

QUESTION 1 of 10

We already know that Pythagoras was a top bloke.

He did come up with the theorem on how to work out the length of sides for triangles after all.

Let us put our thinking cap on the revise the theorem

Remember this theorem only works for right angled triangles.

To find the hypotenuse (longer side) of a triangle

10² + 3² =  √109 = 10.44 cm to 2 decimal places

To find the shorter side length

9² - 7² =√32 = 5.66 to 2 decimal places

The key point here is to remember when to add and when to subtract.

Now we are okay with using Pythagoras' theorem, we can apply this to solve all sorts of problems.

IF IN DOUBT, SKETCH IT OUT

I leave my house to go to the bus stop.

I walk 20 m due West and then walk another 30 m due North.

If I had taken the direct route how far would I have walked.

30 M___________

20 M

Here you can see a right angled triangle has been formed.

The direct route would the the diagonal (hypotenuse)

20² + 30² = √1300 = 36.05 m

You are always looking to sketch the situation out, or turn a diagram into a right angles triangle.

How long is the diagonal of this sponge if the length is 10 cm and the width is 4 cm?

This boat is taking tourist on a local cruise.

It leaves port and travels due west for 15 km and then due south for another 14 km to a different port.

How far would it have traveled if it had gone direct from port to port?

Answers have been rounded to 2 decimal places (2 d.p)

14.8 km

21.65 km

14.5 km

20.52 km

A ladder 10 m long is leaning against the wall of a house.

The ladder reaches 8 m up the wall of the house.

How far away from the base of the wall is the ladder?

14.8 km

21.65 km

14.5 km

20.52 km

An aircraft leaves Bristol airport and heads due east for 120 km, before turning due south for a further 200 km before landing.

On the return journey the pilot flies back direct to Bristol airport.

What is the distance of the return flight.

What is the difference in distance?

 243.50 km 320.00 km 233.24 km 76.86 km 86.76 km 90.21 km Return flight Difference in distance

Your cat is stuck up a tree.

You put up a ladder of length 5 m to go and rescue it.

The base of the ladder is 2 m away from the tree.

As you get to the top of the tree, your cat decides to run down the tree trunk to safety.

How far did the cat run?

George takes a rectangular piece of material and cuts 14 cm along the diagonal.

The width of the material is  7 cm.

What is the length of the material?

14.70 cm

12.12 cm

14.24 cm

11.06 cm

A blackbird has spotted a juicy worm for lunch.

The bird is in a tree that is 12 m high.

The worm is 14 m away from the base of the tree.

How far does the blackbird have to swoop to catch his lunch?

14.70 cm

12.12 cm

14.24 cm

11.06 cm

The ladder on the slide is 1.5 m high, the slide itself is 4 m long.

How far away from the base of the steps does the slide reach.

There are four flamingos in each corner of a rectangular lake.

Two flamingos at the top of the lake want to walk along the edge of the lake to be with their mate.

One side of the lake measures 20 m and the diagonal across measures 25 m.

How far do the flamingos have to walk?

 12m 14m 15 m 18 m Width of the lake is

Daniel is standing looking at this statue.

The statue is 10 m taller than Daniel who is standing 12 m away.

How far is the top of the statue from the top of Daniels head?

• Question 1

How long is the diagonal of this sponge if the length is 10 cm and the width is 4 cm?

10.77
EDDIE SAYS
Just sketch it out and draw a line through the diagonal to see the right angles triangle. Here we can see we need the hypotenuse so it is adding. 10² + 4² = √116 = 10.77 to 2 decimal places
• Question 2

This boat is taking tourist on a local cruise.

It leaves port and travels due west for 15 km and then due south for another 14 km to a different port.

How far would it have traveled if it had gone direct from port to port?

Answers have been rounded to 2 decimal places (2 d.p)

20.52 km
EDDIE SAYS
Did you sketch this out? I always find it easier to do so. Your sketch should show a right angled triangle with the hypotenuse to find. 15² + 14² =√ 421 = 20.52 km to 2 d.p. It clearly can't be option a or c as the lengths are shorter than the hypotenuse.
• Question 3

A ladder 10 m long is leaning against the wall of a house.

The ladder reaches 8 m up the wall of the house.

How far away from the base of the wall is the ladder?

EDDIE SAYS
Your sketch should have shown you that this time you wanted a shorter side of the triangle. 10² - 8² = √36 = 6 m
• Question 4

An aircraft leaves Bristol airport and heads due east for 120 km, before turning due south for a further 200 km before landing.

On the return journey the pilot flies back direct to Bristol airport.

What is the distance of the return flight.

What is the difference in distance?

 243.50 km 320.00 km 233.24 km 76.86 km 86.76 km 90.21 km Return flight Difference in distance
EDDIE SAYS
The sketch will have given you a hypotenuse to find. So it is the adding we want again. 120² + 200² = &radaic;54400 = 233.24 Now add up the distances for the first journey 200+ 120 = 320 now subtract 233.24 to get the direct distance. 320 - 233.24 = 86.76 km I bet the pilot was pleased he could get home for his tea early.
• Question 5

Your cat is stuck up a tree.

You put up a ladder of length 5 m to go and rescue it.

The base of the ladder is 2 m away from the tree.

As you get to the top of the tree, your cat decides to run down the tree trunk to safety.

How far did the cat run?

4.58
EDDIE SAYS
Don't you just love it when cats do that? You were looking for a shorter side of the triangle here so it was a subtraction. 5² - 2² = √21 = 4.58 cm to 2 d.p I hope you didn't get stuck up there instead of the cat.
• Question 6

George takes a rectangular piece of material and cuts 14 cm along the diagonal.

The width of the material is  7 cm.

What is the length of the material?

12.12 cm
EDDIE SAYS
Can you see how sketching things out makes it easier for us to spot whether we need to add or subtract? 14² - 7² = √147 = 12. 12 cm It clearly can't be option a or c as the lengths are longer than the hypotenuse.
• Question 7

A blackbird has spotted a juicy worm for lunch.

The bird is in a tree that is 12 m high.

The worm is 14 m away from the base of the tree.

How far does the blackbird have to swoop to catch his lunch?

EDDIE SAYS
Do you think the bird won, or did the worm get away? Pretty even I think. The sketch shows you need to find the hypotenuse so this is an add. 12² + 14² = √ 340 = 18.44 to 2 d.p
• Question 8

The ladder on the slide is 1.5 m high, the slide itself is 4 m long.

How far away from the base of the steps does the slide reach.

3.7
EDDIE SAYS
Are you getting used to when to add and when to take away? This is the part where students generally make the mistake, so it is always worth doing that sketch. 4² - 1.5² = √13.75 = 3.7 to 1 d.p
• Question 9

There are four flamingos in each corner of a rectangular lake.

Two flamingos at the top of the lake want to walk along the edge of the lake to be with their mate.

One side of the lake measures 20 m and the diagonal across measures 25 m.

How far do the flamingos have to walk?

 12m 14m 15 m 18 m Width of the lake is
EDDIE SAYS
This sketch should give you a shorter side to find. 25² - 20² = √ 225 cm² = 15 cm I love flamingos, but don't think I could stand on one leg as long as they do. I wonder why they do that?
• Question 10

Daniel is standing looking at this statue.

The statue is 10 m taller than Daniel who is standing 12 m away.

How far is the top of the statue from the top of Daniels head?

15.62 m
EDDIE SAYS
This sounds more complicated than it is, as you see when you sketch it out it is just another right angled triangle. 10² + 12² = √244 cm² = 15.62 m I wonder who the statue is..
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