Pythagoras’ theorem Worked Solutions — GCSE Maths

Fully worked, step-by-step solutions to GCSE Pythagoras’ theorem questions. See exactly how to solve problems on Pythagoras' theorem, finding the hypotenuse, finding a shorter side, rounding to a given accuracy.

Pythagoras' theoremfinding the hypotenusefinding a shorter siderounding to a given accuracydistance between two pointscoordinates
GCSE Foundation70 questionsStep-by-step solutions
Question 1
2 markseasy
A right-angled triangle has shorter sides of length 33 cm and 44 cm. Work out the length of the hypotenuse.

Worked solution

  1. State Pythagoras' theorem.

    a2+b2=c2a^2 + b^2 = c^2

    In a right-angled triangle the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides: a2+b2=c2a^2 + b^2 = c^2.

  2. Write the theorem out for this triangle.

    c2=32+42c^2 = 3^2 + 4^2

    The hypotenuse cc is the side opposite the right angle; the two shorter sides are 33 cm and 44 cm.

  3. State the length of the hypotenuse.

    c=9+16=25=5 cmc = \sqrt{9 + 16} = \sqrt{25} = 5\text{ cm}

    The hypotenuse is 55 cm.

Answer
c=5 cmc = 5\text{ cm}
Question 2
2 markseasy
A right-angled triangle has shorter sides of length 66 cm and 88 cm. Work out the length of the hypotenuse.

Worked solution

  1. State Pythagoras' theorem.

    a2+b2=c2a^2 + b^2 = c^2

    In a right-angled triangle the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides: a2+b2=c2a^2 + b^2 = c^2.

  2. Write the theorem out for this triangle.

    c2=62+82c^2 = 6^2 + 8^2

    The hypotenuse cc is the side opposite the right angle; the two shorter sides are 66 cm and 88 cm.

  3. State the length of the hypotenuse.

    c=36+64=100=10 cmc = \sqrt{36 + 64} = \sqrt{100} = 10\text{ cm}

    The hypotenuse is 1010 cm.

Answer
c=10 cmc = 10\text{ cm}
Question 3
2 markseasy
A right-angled triangle has shorter sides of length 55 cm and 1212 cm. Work out the length of the hypotenuse.

Worked solution

  1. State Pythagoras' theorem.

    a2+b2=c2a^2 + b^2 = c^2

    In a right-angled triangle the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides: a2+b2=c2a^2 + b^2 = c^2.

  2. Write the theorem out for this triangle.

    c2=52+122c^2 = 5^2 + 12^2

    The hypotenuse cc is the side opposite the right angle; the two shorter sides are 55 cm and 1212 cm.

  3. State the length of the hypotenuse.

    c=25+144=169=13 cmc = \sqrt{25 + 144} = \sqrt{169} = 13\text{ cm}

    The hypotenuse is 1313 cm.

Answer
c=13 cmc = 13\text{ cm}
Question 4
2 markseasy
A right-angled triangle has shorter sides of length 88 m and 1515 m. Work out the length of the hypotenuse.

Worked solution

  1. State Pythagoras' theorem.

    a2+b2=c2a^2 + b^2 = c^2

    In a right-angled triangle the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides: a2+b2=c2a^2 + b^2 = c^2.

  2. Write the theorem out for this triangle.

    c2=82+152c^2 = 8^2 + 15^2

    The hypotenuse cc is the side opposite the right angle; the two shorter sides are 88 m and 1515 m.

  3. State the length of the hypotenuse.

    c=64+225=289=17 mc = \sqrt{64 + 225} = \sqrt{289} = 17\text{ m}

    The hypotenuse is 1717 m.

Answer
c=17 mc = 17\text{ m}
Question 5
2 markseasy
A right-angled triangle has a hypotenuse of length 1313 cm and one shorter side of length 55 cm. Work out the length of the other shorter side.

Worked solution

  1. State Pythagoras' theorem.

    a2+b2=c2a^2 + b^2 = c^2

    In a right-angled triangle the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides: a2+b2=c2a^2 + b^2 = c^2.

  2. Write the theorem out for this triangle and rearrange it.

    52+x2=132⇒x2=132−525^2 + x^2 = 13^2 \Rightarrow x^2 = 13^2 - 5^2

    The hypotenuse is 1313 cm, so it goes on its own on one side of the theorem. The missing side is one of the SHORTER sides, so subtract.

  3. State the length of the missing side.

    x=169−25=144=12 cmx = \sqrt{169 - 25} = \sqrt{144} = 12\text{ cm}

    The other shorter side is 1212 cm.

Answer
x=12 cmx = 12\text{ cm}

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