# The Correct Angle is the Right One

In my last post, you saw that a 90° angle is called a right angle. This is the angle made by the two lines at the corner of a square. Now a triangle is a shape that has three angles inside. A basic property of any triangle is that all the internal angles add up to 180°:

But this post is about triangles where one of its angles is 90°, that is a right angle. Such triangles are called right triangles.

Below is a right triangle where I have labelled the sides as a, b, and c. Side c is the side opposite the right angle. This side is called the hypotenuse. The hypotenuse is always the longest side of any right triangle.

Right triangles have another famous property that relates the lengths of the three sides. This property is called the Pythagorus Theorem. This is named after the Greek mathematician Pythagorus who lived 570 to 495 BCE. This theorem was used before his time but he is credited with providing the first proof. Given the sides as labelled above, the following is true for any size right triangle:

c² = a² + b²

This means that if you know any two sides of a right triangle, you can calculate the third side using this equation. Let’s do an example:

So we now know that c² = 4² + 3² = 16 + 9 = 25. You can now find c by taking the square root of both side of the equation. Square roots have been covered in previous posts:

${c}^{2}\hspace{0.33em}{=}\hspace{0.33em}{25}\hspace{0.33em}\hspace{0.33em}\Longrightarrow\hspace{0.33em}\hspace{0.33em}\sqrt{{c}^{2}}\hspace{0.33em}{=}\hspace{0.33em}\sqrt{25}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\Longrightarrow\hspace{0.33em}\hspace{0.33em}{c}\hspace{0.33em}{=}\hspace{0.33em}{5}$

In my posts on square roots, I did say that taking a square root results in two solutions, one positive and one negative. But since we are solving for a physical length, we can ignore the negative solution. So the hypotenuse for this triangle is 5.

Not all right triangle problems work out so well. Most square roots are decimal numbers and you have to either round the number or leave the answer as a square root.

The Pythagorus Theorem can also be used to find one of the non-hypotenuse sides as well:

You can rearrange the theorem’s equation to solve for the unknown side:

$\begin{array}{l}{{c}^{2}\hspace{0.33em}{=}\hspace{0.33em}{a}^{2}\hspace{0.33em}{+}\hspace{0.33em}{b}^{2}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\Longrightarrow\hspace{0.33em}\hspace{0.33em}{b}^{2}\hspace{0.33em}{=}\hspace{0.33em}{c}^{2}\hspace{0.33em}{-}\hspace{0.33em}{a}^{2}}\\{{b}^{2}\hspace{0.33em}{=}\hspace{0.33em}{9}^{2}\hspace{0.33em}{-}\hspace{0.33em}{5}^{2}\hspace{0.33em}{=}\hspace{0.33em}{81}\hspace{0.33em}{-}\hspace{0.33em}{25}\hspace{0.33em}{=}\hspace{0.33em}{56}}\\{\sqrt{{b}^{2}}\hspace{0.33em}{=}\hspace{0.33em}\sqrt{56}\hspace{0.33em}\hspace{0.33em}\Longrightarrow\hspace{0.33em}\hspace{0.33em}{b}\hspace{0.33em}\approx\hspace{0.33em}{7}{.}{48}}\end{array}$

So b approximately equals 7.48. That is what the symbol “≈” means. The exact answer cannot be written as a decimal number as the decimal part goes on forever.

Posted on Categories Pre-VCE, Trigonometry