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GCSE Maths · Higher · Chapter 11: Right-angled triangles · Lesson 1

Pythagoras' theorem

Grades 4–7About 40 minutesCalculator allowed34 marks
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NameClassDate
A

Do Now

Grades 4–6

The first question is what today's lesson needs. The others bring back earlier topics.

  1. A1
    Needed today

    Work out .

    [1 mark]
    Answer
  2. A2
    From chapter 9

    A cylinder has radius 4 cm and height 10 cm. Find its volume in terms of .

    [1 mark]
    Answer
  3. A3
    Last lesson

    A plumber charges pounds for hours. What is the fixed charge?

    [1 mark]
    Answer
  4. A4
    Key Stage 3

    Work out .

    [1 mark]
    Answer
B

Example, then your turn

Grades 4–6

Your teacher works through each example with you. Then try the one beside it.

Example 1

Work out the hypotenuse, . Give your answer to 1 decimal place.

9.4 cm6.3 cmx
Not drawn accurately
Hint Square the two shorter sides, add, then take the square root.
  1. B1

    Work out the hypotenuse, . Give your answer to 1 decimal place.

    [3 marks]
    11.5 cm7.2 cmx
    Not drawn accurately
    Answer
Example 2

Work out the length . Give your answer to 1 decimal place.

x8.9 cm15.6 cm
Not drawn accurately
Hint The hypotenuse is the longest side, so subtract.
  1. B2

    Work out the length . Give your answer to 1 decimal place.

    [3 marks]
    x12.4 cm19.3 cm
    Not drawn accurately
    Answer
C

Practice

Grades 5–6
  1. C1

    A triangle has sides of 48 cm, 55 cm and 73 cm. Is it right-angled? Show how you decide.

    [2 marks]
  2. C2

    Work out the length . Give your answer to 1 decimal place.

    [2 marks]
    6.5 cm14.2 cmx
    Not drawn accurately
    Answer
  1. C3

    The triangle is isosceles. Work out its height, .

    [2 marks]
    16 cm17 cm17 cmh
    Not drawn accurately
    Answer
  2. C4

    A rectangular gate is 3.6 m long and 1.5 m high. A diagonal brace runs from one corner to the opposite corner. Work out the length of the brace.

    [2 marks]
    3.6 m1.5 m
    Not drawn accurately
    Answer
  3. C5

    is the point and is the point . Work out the length of . Give your answer to 1 decimal place.

    [3 marks]
    xyA(−2, 5)B(7, −1)O
    Not drawn accurately
    Answer
D

Exam-style questions

Grades 6–7
  1. D1

    A right-angled triangle has a hypotenuse of 13.6 cm and another side of 6.4 cm. Ravi says, "The third side is cm."

    Explain why Ravi is wrong, and work out the third side.

    [2 marks]
  1. D2

    The cross-section of a roof is an isosceles triangle. Its base is 9.6 m wide and each sloping edge is 6.2 m long.

    (a)

    Work out the height of the roof. Give your answer to 1 decimal place.

    [3 marks]
    Answer
    (b)

    Work out the area of the cross-section. Give your answer to 3 significant figures.

    [2 marks]
    Answer
    (c)

    Is the angle at the top of the roof bigger or smaller than ? Use the side lengths to decide.

    [2 marks]
    Total for D2: 7 marks
E

Extension

Grade 8 and beyond

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

  1. E1

    The sides of a right-angled triangle are cm, cm and cm. Work out the value of .

    [4 marks]
    Hint 1 · What to try

    The longest side, , is the hypotenuse. Write down Pythagoras with these three lengths.

    Hint 2 · The first line

    .

    Hint 3 · The full method

    Expand: , so , and , as a length cannot be negative.

    Answer
F

How did it go?

Colour one circle on each line.
Not yetNearlyYes
I can find the hypotenuse or a shorter side of a right-angled triangle.
I can test whether a triangle has a right angle.
I can use Pythagoras in isosceles triangles, rectangles and on a coordinate grid.

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