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3.02 Finding the hypotenuse

Lesson

The hypotenuse

Pythagoras' theorem states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. The theorem can be written algebraically: a^{2}+b^{2}=c^{2}

where c represents the length of the hypotenuse and a and b are the two shorter sides.

A right angled triangle with sides a, b, and c on the hypotenuse. Pythagoras theorem is written next to it.

We can use the formula to find the length of the hypotenuse of any right-angled triangle, as long as we know the lengths of the other two sides.

In some cases, where the sides of the right-angled triangle form a Pythagorean triad, the exact length of the hypotenuse is an integer value. But for most cases we will end up with an irrational number, that is, a surd.

If we are asked to give an exact answer, or to answer as a surd, we can stop our working out when we arrive at a line of working such as c=\sqrt{11}, as this can not be simplified without losing some accuracy when it is rounded.

Examples

Example 1

Find the length of the unknown side c in the triangle below. Give the answer as a surd.

A right angled triangle with 2 short side lengths of 11 and 13, and the longest side length of c.
Worked Solution
Create a strategy

We can use the Pythagoras' theorem: a^{2}+b^{2} = c^{2}.

Apply the idea
\displaystyle c^{2}\displaystyle =\displaystyle a^{2}+b^{2}Write the formula
\displaystyle c^{2}\displaystyle =\displaystyle 11^{2}+13^{2}Substitute a and b
\displaystyle c^{2}\displaystyle =\displaystyle 121+169Evaluate the squares
\displaystyle c^{2}\displaystyle =\displaystyle 290Evaluate the sum
\displaystyle c\displaystyle =\displaystyle \sqrt{290}Square root both sides
Reflect and check

The formula gives us a hypotenuse length of \sqrt{290} \approx 17.0294. This is larger than both our shorter sides, as we should expect. If we got a number smaller than one (or both) of the short sides we know we have made a mistake in our calculation.

Idea summary

Pythagoras' theorem:

\displaystyle a^2+b^2=c^2
\bm{c}
is the length of the hypotenuse
\bm{a}
is the length of a shorter side
\bm{b}
is the length of the other shorter side

Outcomes

VCMMG318

Investigate Pythagoras’ Theorem and its application to solving simple problems involving right-angled triangles.

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