Pythagoras' theorem

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Apuntes de la lección

The Hypotenuse

  • In a right-angled triangle, the hypotenuse is the side opposite the right angle.
  • The hypotenuse is always the longest side of a right-angled triangle.
  • The other two sides are sometimes called the legs or the shorter sides.
  • The right angle is 90° and is usually marked with a small square.

Pythagoras' Theorem

  • Pythagoras' theorem states that for any right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.
  • If the two shorter sides have lengths a and b, and the hypotenuse has length c, then a2 + b2 = c2.
  • This relationship only holds for right-angled triangles.
  • The theorem is named after the Greek philosopher Pythagoras, who was born around 570 BC.
  • Many different proofs of the theorem exist, using both geometry and algebra.

Finding the Hypotenuse

  • To find the hypotenuse, square the two shorter sides, add the results, then take the square root.
  • Example: if a = 3 and b = 4, then c2 = 32 + 42 = 9 + 16 = 25, so c = √25 = 5.
  • The answer for a length should be given as a positive number.
  • If the square root is not exact, round to a sensible number of decimal places, such as 2 decimal places.

Finding a Shorter Side

  • To find one of the shorter sides, rearrange the formula: a2 = c2 - b2 or b2 = c2 - a2.
  • Subtract the square of the known shorter side from the square of the hypotenuse.
  • Then take the square root of the result.
  • Example: if c = 13 and a = 5, then b2 = 132 - 52 = 169 - 25 = 144, so b = √144 = 12.

Deciding if a Triangle is Right-Angled

  • To test whether a triangle is right-angled, check whether the square of the longest side equals the sum of the squares of the other two sides.
  • If a2 + b2 = c2 (where c is the longest side), the triangle is right-angled.
  • If a2 + b2 > c2, the triangle is acute-angled (all angles less than 90°).
  • If a2 + b2 < c2, the triangle is obtuse-angled (one angle greater than 90°).
  • Example: a triangle with sides 6, 8, 10 is right-angled because 62 + 82 = 36 + 64 = 100 = 102.

Pythagorean Triples

  • A Pythagorean triple is a set of three positive integers that satisfy a2 + b2 = c2.
  • The most common triple is 3, 4, 5.
  • Other examples include 5, 12, 13; 8, 15, 17; and 7, 24, 25.
  • Multiples of a triple are also triples: for example, 6, 8, 10 is 2 × (3, 4, 5).
  • Pythagorean triples have been known since ancient times, with examples in Babylonian and Indian texts.

Real-Life Applications

  • Pythagoras' theorem can be used to find distances that cannot be measured directly, such as the height of a ladder against a wall.
  • It can also be used to check whether a corner is a right angle, for example in building or carpentry.
  • When solving a real-life problem, sketch the situation and identify the right-angled triangle.
  • Label the sides clearly with the given lengths and the unknown side.
  • Remember to include units in your final answer.

Distance Between Two Coordinates

  • The distance between two points (x1, y1) and (x2, y2) can be found using Pythagoras' theorem.
  • The horizontal difference is x2 - x1 and the vertical difference is y2 - y1.
  • The distance d satisfies d2 = (x2 - x1)2 + (y2 - y1)2.
  • So d = \√{(x2 - x1)2 + (y2 - y1)2}.
  • Example: the distance between (1, 2) and (4, 6) is \√{(4-1)2 + (6-2)2} = \√{9 + 16} = \√{25} = 5 units.

Diapositivas

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Preguntas de práctica

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  1. 1.In a right-angled triangle, what is the name of the side opposite the right angle?

    Easy
    • Ahypotenuse
    • Badjacent
    • Copposite
    • Dperpendicular

  2. 2.Which equation correctly states Pythagoras' theorem for a right-angled triangle with legs a and b and hypotenuse c?

    Easy
    • Aa2 + b2 = c2
    • Ba + b = c
    • Ca2 - b2 = c2
    • Da2 + c2 = b2

  3. 3.A right-angled triangle has legs of length 3 cm and 4 cm. What is the length of the hypotenuse?

    Medium
    • A5 cm
    • B7 cm
    • C12 cm
    • D25 cm

  4. 4.Pythagoras' theorem can only be used in right-angled triangles.

    Easy

    True or false?

  5. 5.Which of the following sets of three numbers is a Pythagorean triple?

    Medium
    • A6, 8, 10
    • B1, 2, 3
    • C5, 12, 14
    • D8, 15, 16

  6. 6.Which of the following are Pythagorean triples? (select all that apply)

    Medium
    • A3, 4, 5
    • B5, 12, 13
    • C1, 1, 2
    • D7, 24, 25
    • E2, 3, 4

  7. 7.Arrange the steps to find the hypotenuse c of a right-angled triangle with legs a and b in the correct order.

    Medium
    • Write down a2 + b2 = c2
    • Substitute the values of a and b
    • Square a and b and add the results
    • Take the square root to find c

  8. 8.Match each description to the correct side of a right-angled triangle.

    Easy
    • the side opposite the right angle
    • the longest side
    • the side next to the right angle
    • hypotenuse
    • leg

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