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12.2 Pythagoras' Theorem 2

GCSE Maths Foundation · Edexcel · Chapter 12: Right-angled triangles

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: finding a shorter side, and solving problems with Pythagoras’ theorem. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with interactive activities, practice quizzes and printable worksheets, in our free online course.

What you will learn

  • Rearrange Pythagoras’ theorem to find one of the shorter sides
  • Decide whether to add or subtract the squares
  • Solve problems involving ladders, ramps, rectangles and isosceles triangles
  • Use Pythagoras’ theorem twice in problems with two triangles

Key words

  • Hypotenuse: The longest side, opposite the right angle.
  • Shorter side: Either of the two sides that make the right angle.
  • Isosceles triangle: A triangle with two equal sides. Its height cuts the base into two equal halves.
  • Perpendicular height: The height measured at right angles to the base.

Finding a shorter side

If you know the hypotenuse and one shorter side, subtract: a² = c² − b². Square the hypotenuse, take away the square of the other side, then square root.

Worked example: rounding

A right-angled triangle has hypotenuse 9.5 cm and one side 6 cm. Find the third side to 1 d.p.
a² = 9.5² − 6² = 90.25 − 36 = 54.25
a = √54.25 = 7.365… = 7.4 cm

Answer: 7.4 cm

Add or subtract?

First find the hypotenuse on the diagram. Then decide.

  • the hypotenuse; Do this add the squares, then √; Check answer is longer than both sides
  • a shorter side; Do this subtract the squares (big − small), then √; Check answer is shorter than the hypotenuse

Worked example: checking the answer

A side of 8 cm and a hypotenuse of 17 cm. Find the third side.
17² − 8² = 289 − 64 = 225, √225 = 15. 15 < 17, so it makes sense.

Answer: 15 cm

Ladders, ramps and rectangles

A ladder against a wall makes a right-angled triangle: the ladder is the hypotenuse. A rectangle’s diagonal makes two right-angled triangles.

Worked example: a ladder

A 5 m ladder leans against a wall with its foot 1.4 m from the wall. How high up the wall does it reach?
h² = 5² − 1.4² = 25 − 1.96 = 23.04
h = √23.04 = 4.8

Answer: 4.8 m

Worked example: a rectangle

A rectangle has a diagonal of 17 cm and a width of 8 cm. Find its area.
Length² = 17² − 8² = 225, so length = 15 cm. Area = 15 × 8 = 120 cm²

Answer: 120 cm²

Isosceles triangles and two-step problems

The height of an isosceles triangle splits it into two identical right-angled triangles. Use half the base. In a two-step problem, find the side the two triangles share first.

Worked example: area of an isosceles triangle

An isosceles triangle has sides 10 cm, 10 cm and 12 cm. Find its area.
h² = 10² − 6² = 64, so h = 8 cm.
Area = ½ × 12 × 8 = 48 cm²

Answer: 48 cm²

Common mistakes

  • Adding the squares when finding a shorter side. If you know the hypotenuse, subtract.
  • Subtracting the wrong way round: work out (hypotenuse)² − (side)², not the other way.
  • Using the full base of an isosceles triangle instead of half of it.
  • Giving an answer for a shorter side that is longer than the hypotenuse. Always check.

Revision questions

Try these, then check your answers below each one.

Question 1. A right-angled triangle has hypotenuse 25 cm and one side 7 cm. Work out the third side.

Answer: 24 cm

Working: 25² - 7² = 625 - 49 = 576. √576 = 24 cm.

Question 2. A right-angled triangle has hypotenuse 10 cm and one side 4 cm. Work out the third side, correct to 1 decimal place.

Answer: 9.2 cm

Working: 10² - 4² = 84. √84 = 9.165… ≈ 9.2 cm.

Question 3. A right-angled triangle has hypotenuse 13 cm and one side 5 cm. Which calculation gives the third side? (Choose from: √(13² + 5²), √(13² - 5²), 13 - 5, √(5² - 13²).)

Answer: √(13² - 5²)

Working: √(169 - 25) = √144 = 12 cm.

Related lessons

Continue in the full course

This lesson is section 12.2 (EGCMF1202) of our free GCSE Maths Foundation course. In the full course you get every video for this section, an animated interactive lesson, a self-marking practice quiz, a printable worksheet with a mark scheme, and your progress saved. Go to the full GCSE Maths Foundation course.

 

 

 

 

 

 

 

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