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2. Use the Theorem of Pythagoras to work out the length of the unknown sides.

(a) A right triangle with sides 2m and 6m, and hypotenuse labeled 'a'.

(b) A right triangle with sides 3cm and 3cm, and hypotenuse labeled 'd'.

(c) A right triangle with sides 8cm and 17cm, and hypotenuse labeled 'c'.

(d) A right triangle with sides 35mm and 12mm, and hypotenuse labeled 'd'.

Answer :

To solve these problems, we'll use the Pythagorean Theorem, which states that in a right triangle, the square of the length of the hypotenuse [tex]c[/tex] is equal to the sum of the squares of the lengths of the other two sides [tex]a[/tex] and [tex]b[/tex]. The theorem is expressed as:

[tex]c^2 = a^2 + b^2[/tex]

Let's apply this theorem to the given problems:

  1. (a) Right triangle with sides 2m and 6m, and hypotenuse labeled 'a'.

    Here, the sides are [tex]2\, \text{m}[/tex] and [tex]6\, \text{m}[/tex]. Using the Pythagorean Theorem:

    [tex]a^2 = 2^2 + 6^2[/tex]
    [tex]a^2 = 4 + 36[/tex]
    [tex]a^2 = 40[/tex]
    [tex]a = \sqrt{40} = 2\sqrt{10} \approx 6.32\, \text{m}[/tex]

  2. (b) Right triangle with sides 3cm and 3cm, and hypotenuse labeled 'd'.

    The sides are both [tex]3\, \text{cm}[/tex]. Using the Pythagorean Theorem:

    [tex]d^2 = 3^2 + 3^2[/tex]
    [tex]d^2 = 9 + 9[/tex]
    [tex]d^2 = 18[/tex]
    [tex]d = \sqrt{18} = 3\sqrt{2} \approx 4.24\, \text{cm}[/tex]

  3. (c) Right triangle with sides 8cm and 17cm, and hypotenuse labeled 'c'.

    Assuming the hypotenuse [tex]c[/tex] is the side labeled 'c', we verify this as follows using the possible configuration:

    [tex]c^2 = 8^2 + 17^2[/tex]
    [tex]c^2 = 64 + 289[/tex]
    [tex]c^2 = 353[/tex]
    [tex]c = \sqrt{353} \approx 18.79\, \text{cm}[/tex]

  4. (d) Right triangle with sides 35mm and 12mm, and hypotenuse labeled 'd'.

    Let's use the Pythagorean Theorem to find [tex]d[/tex]:

    [tex]d^2 = 35^2 + 12^2[/tex]
    [tex]d^2 = 1225 + 144[/tex]
    [tex]d^2 = 1369[/tex]
    [tex]d = \sqrt{1369} = 37\, \text{mm}[/tex]

These calculations show how to apply the Pythagorean Theorem to find unknown sides in right triangles. Always ensure the side labeled as the hypotenuse is opposite the right angle.

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