Video: Finding the Length of the Hypotenuse in a Right-Angled Triangle Using Right-Angled Triangle Trigonometry

Find the length of 𝐴𝐶.

02:43

Video Transcript

Find the length of 𝐴𝐶.

From the diagram, we can see that we have a right-angled triangle in which we know the length of one side, 7.5 centimeters, and the size of one of the other angles, 30 degrees. Incidentally, we do also know the size of the third angle in this triangle as the angle sum in a triangle is fixed to 180 degrees.

We’ve been asked to calculate the length of one of the other sides. In order to do this, we need to apply some trigonometry. Trigonometry uses the fact that the ratios between the different pairs of sides of a right-angled triangle are always fixed for a given angle, in this case 30 degrees.

Let’s begin by labelling the three sides of this triangle in relation to the angle of 30 degrees. The longest side, the side opposite the right angle, is called the hypotenuse; the side opposite the other known angle, in this case 30 degrees, is called the opposite; the third side that is the side between the right angle and the known angle is called the adjacent. The two sides that are going to be involved in the ratio in this question are the side we know, which is the opposite side, and the side we want to calculate, which is the hypotenuse.

We need to recall a key fact about the ratio between the opposite and the hypotenuse in a right-angled triangle when the angle that we’ve been given is 30 degrees. And it’s this; that ratio of the opposite divided by the hypotenuse is always equal to a half. Remember this isn’t true for any angle at all, but it is true when the angle that the sides have been labelled in relation to is 30 degrees, as it is here.

If the ratio of the opposite divided by the hypotenuse is a half, then this means the hypotenuse is twice the length of the opposite, and you can see this by cross multiplying. So in this triangle, we know the length of the opposite and we want to calculate the length of the hypotenuse. Therefore, all we need to do is double it.

So we have the length of 𝐴𝐶 is equal to two multiplied by the length of 𝐴𝐵 that is two multiplied by 7.5, and therefore, the length of 𝐴𝐶 is 15 centimeters. Remember we answered this question by recalling that the ratio between the opposite and the hypotenuse in a right-angled triangle is always a half if the angle that they’re labelled in relation to is 30 degrees.

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