 # Solve Complex problems Using Pythagoras' Theorem

In this worksheet, students will apply Pythagoras' theorem to solve more problems. Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   Eduqas, AQA, OCR, Pearson Edexcel

Curriculum topic:   Geometry and Measures, Mensuration

Curriculum subtopic:   Mensuration and Calculation, Triangle Mensuration

Difficulty level:   ### QUESTION 1 of 10 We already know that our friend Pythagoras here, had the brain the size of a planet.

He must have as it is said that he never wrote any of his theories down. 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.

We can apply this to solve all sorts of problems. However they may be presented.

Question 1

The diagonal of a square is 10 cm long.  What is the length of one side of the square? IF IN DOUBT, SKETCH IT OUT Now don't be tempted to think because this is a square it will be 10 cm.

Pythagoras gave us his theorem for a reason.

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

We know that ?² + ?² = 10² (100 cm)

The sides will both be the same length as we are looking at a square.

100 ÷ 2 = 50 cm² √ 50 cm² = 7.07 cm rounded to 2 decimal places

Question 2   You are always looking to sketch the situation out to turn a diagram into a right angles triangle.

In more complex problems you will need to draw on other knowledge too. Find the value of x

7.2 cm

9.5 cm

18 cm

4.24 cm A speed boat races 20 km due North and then 35 Km due West.

How far is the boat from the starting point? 6.32 cm 6.50 cm 42.32 cm 40 cm The value of x is A cone has base radius of 5 cm and a slant height of 11 cm.

What is the vertical height?

Answers have been rounded to 2 decimal places.

9.80 cm

9.79 cm

9.64 cm

9.58 cm The value of x is 5.02 cm, 4.20 cm, 5.66 cm, 4.65 cm The value of x is 5.02 cm, 4.20 cm, 5.66 cm, 4.65 cm A jet plane flies equal distances North West and then North East to finish 120 km due North of its starting point.  How long is each leg of the journey? The diagonal of a rectangle exceeds the length by 2 cm. If the width of the rectangle is 10 cm, find the length. A 25 m ladder leans against a vertical wall. The foot of the ladder is 18 m away from the wall. If the foot is moved 7 m closer to the wall.  How far up the wall does the ladder now reach?

Answers have been rounded to 1 decimal place.

7.1 cm

6.1 cm

5.1 cm

4.1 cm AG is 14 cm

EG is 11 cm

Calculate the length of  AE

5.45 cm

6.54 cm

8.66 cm

7.54 cm

• Question 1 Find the value of x

4.24 cm
EDDIE SAYS
Don't give up on questions like this. You have all the knowledge that you need. Just apply what you know Here we know that 6² = 36 cm We need to share this 36 cm between the two x's. x = 18 cm² We want a side length so √18 = 4.24 cm
• Question 2 A speed boat races 20 km due North and then 35 Km due West.

How far is the boat from the starting point?

40.31
EDDIE SAYS
Did you sketch out the problem? This often helps to visualise things more. You can see straight away that it is a right angled triangle which means Pythagoras. 20² + 35² = 1625 km² √1625 = 40.31 km
• Question 3 6.32 cm 6.50 cm 42.32 cm 40 cm The value of x is
EDDIE SAYS
You love it so much, you did it twice.!! In problem solving you sometimes have to apply Pythagoras Theorem twice. But you have got this. 7² - 3² = 40cm² √ 40 = 6.32 cm Now we can find the value of x. 6.32² + 6² = 42.32 cm² √ 42.32 = 6.50 cm
• Question 4 A cone has base radius of 5 cm and a slant height of 11 cm.

What is the vertical height?

Answers have been rounded to 2 decimal places.

9.80 cm
EDDIE SAYS
11² - 5² = 96 cm² √96 = 9.80 cm correct to 2 decimal places. Were you careful with the rounding here?
• Question 5 The value of x is 5.02 cm, 4.20 cm, 5.66 cm, 4.65 cm
EDDIE SAYS
This is where you need to draw on other knowledge. Before we can do this we have to find the vertical height first. 5² - 3² = 16 cm² √ 16 = 4 cm. We can see from the markings on the triangle that it is an isosceles triangle, therefore the base required to find x is also 4 cm. 4² + 4² = 32 cm² √32 = 5.66 to 2 decimal places
• Question 6 EDDIE SAYS
Algebra.. no problem for us. 8² + (x - 2)² = x² (x - 2)(x - 2) = x² -2x -2x + 4 Simplified = x² - 4x + 4 8² (64) x² - 4x + 4 = x² - 4x + 68 = x² Solve the equation x² cancels out 4x = 68 x = 17
• Question 7 A jet plane flies equal distances North West and then North East to finish 120 km due North of its starting point.  How long is each leg of the journey?

84.85
EDDIE SAYS
Are you firing on all cylinders now? First find out 120km² = 14400 km² This is now shared between the two legs. 14400 ÷ 2 = 7200 km² √ 7200 = 84.85 km
• Question 8 The diagonal of a rectangle exceeds the length by 2 cm. If the width of the rectangle is 10 cm, find the length.

EDDIE SAYS
Did you sketch this out? If you did you would have labelled the diagonal as x + 2. The length x and the width 10 cm. Use what you know. (x+ 2)² - x² = 10² x² +4x + 4 - x² = 100 Now solve the equation 4x + 4 = 100 4x = 96 x = 24 cm
• Question 9 A 25 m ladder leans against a vertical wall. The foot of the ladder is 18 m away from the wall. If the foot is moved 7 m closer to the wall.  How far up the wall does the ladder now reach?

Answers have been rounded to 1 decimal place.

5.1 cm
EDDIE SAYS
Why can't the ladder stay in one place?? Don't worry we just need to apply Pythagoras Theorem twice. Sketch out what you know. This should lead you to 25² - 18² = 301 m² √301 = 17.34 m Subtract 7 m away from 18 m. This shows that the ladder is now 11 m away from the wall. Apply Pythagoras' theorem again. 25² - 11² = √ 504 = 22.44 m Now just subtract 17.34 m from 22.44 m = 5.1 m
• Question 10 AG is 14 cm

EG is 11 cm

Calculate the length of  AE

8.66 cm
EDDIE SAYS
This is just a case of taking the information from a written format instead of it being on a diagram. 14² - 11² = 75 cm² √ 75 = 8.66 cm Pythagoras..no problem..
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