Given that 𝐴𝐵 is equal to three, find 𝐵𝐶. Give an exact answer.
Now there are two possible approaches that we could take to answering this question. So we’ll look at them both. Firstly, we note that triangle 𝐴𝐵𝐶 is not just a right-angled triangle, but it is in fact an isosceles triangle as it has two equal angles of 45 degrees. In turn, this means that the sides 𝐴𝐵 and 𝐴𝐶 are of equal length. As 𝐴𝐵 is equal to three, this means that 𝐴𝐶 is also equal to three. The first approach that we will consider is to apply the Pythagorean theorem. We can do this because we have a right-angled triangle in which we know the lengths of two sides and we want to calculate the length of the third side.
The Pythagorean theorem tells us that in a right-angled triangle, the sum of the squares of the two shorter sides is equal to the square of the hypotenuse, which is the longest side of a right-angled triangle. The longest side is always the side opposite the right angle. So in this triangle, that’s the side 𝐵𝐶. And so the Pythagorean theorem tells us that 𝐵𝐶 squared is equal to 𝐴𝐵 squared plus 𝐴𝐶 squared. Remember that 𝐴𝐵 and 𝐴𝐶 are each equal to three, so we have that 𝐵𝐶 squared is equal to three squared plus three squared. Three squared is nine. So three squared plus three squared, that’s nine plus nine, is 18. And so we have that 𝐵𝐶 squared is equal to 18.
To find the value of 𝐵𝐶 rather than 𝐵𝐶 squared, we need to take the square root of each side of the equation. And it gives 𝐵𝐶 is equal to the square root of 18. Now we should always simplify surds where possible. And to do so, we need to look for square factors. 18 is equal to nine multiplied by two, and nine is a square number. So we can say that the square root of 18 is equal to the square root of nine multiplied by two. This can be separated into the square root of nine multiplied by the square root of two. And as the square root of nine is just three, the surd simplifies to three root two. So we found then that 𝐵𝐶 is equal to three root two. There are no units for this answer as there were no units given in the original question.
The second method that we can apply is right-angle trigonometry, as we have a right-angled triangle in which we know the length of at least one side and the size of at least one angle and we want to calculate the length of a second side. Let’s begin by labeling the three sides of the triangle in relation to angle 𝐴𝐵𝐶. The hypotenuse of a right-angled triangle is always the same. It’s always the side opposite the right angle. So that’s the side 𝐵𝐶. The opposite is the side opposite the angle that we’re interested in. So the side opposite angle 𝐴𝐵𝐶 is the side 𝐴𝐶. The third side, which is between the angle we know and the right angle, is called the adjacent. So that’s the side 𝐴𝐵. Next, we write down the acronym SOHCAHTOA to help us decide which of the three trigonometric ratios — sine, cosine, or tan — we need to use in this question.
We do actually have a choice here as we know the lengths of both 𝐴𝐵 and 𝐴𝐶. But as it was 𝐴𝐵 that was originally given in the question, we know the adjacent, and we want to calculate the hypotenuse, which means we need to use cos. The definition of the cosine ratio in a right-angled triangle is that cos of the angle 𝜃 is equal to the adjacent divided by the hypotenuse. In our triangle, the angle is 45 degrees, so that’s 𝜃, the adjacent is three units, and the hypotenuse, which we want to calculate, is 𝐵𝐶. We now have an equation that we can solve in order to find the length of 𝐵𝐶. The first step is to multiply both sides of the equation by 𝐵𝐶 as this will eliminate the 𝐵𝐶 in the denominator of the fraction on the right. This gives 𝐵𝐶 multiplied by cos of 45 degrees is equal to three.
The next step is to divide both sides of the equation by cos of 45 degrees, giving 𝐵𝐶 is equal to three divided by cos 45 degrees. Now in all likelihood, you wouldn’t have access to a calculator when answering this question because 45 degrees is a special angle for which the sin, cos, and tan ratios can be written exactly in terms of surds. Cos of 45 degrees is exactly equal to root two over two. You need to learn each of these values. This tells us then that 𝐵𝐶 is equal to three divided by root two over two. However, we need to simplify this answer further. To divide by a fraction, we flip or invert the fraction and multiply. So three divided by root two over two is actually equal to three multiplied by two over root two.
Now two divided by root two is actually just equal to root two. We’re dividing a number by its square root, and so the answer to that calculation will be the square root of the original number. Think about if you divide 16 by its square root, which is four, the answer is also four, the square root of 16. So we have then that 𝐵𝐶 is equal to three multiplied by root two, which we can just write as three root two. This agrees with the answer that we found using the Pythagorean theorem. So we can be confident that our answer of 𝐵𝐶 equals three root two is correct.