Given that the area of a triangle is 52 square centimetres and its base length is 13 centimetres, determine the ratio between its base length and its height.
We’re going to need to be really careful here. We’ve been given the area of a triangle and its base length. Then we’ve been told to find the ratio between its base length and its height. So we’re going to need to do a little bit of work before we can actually write down a ratio. But how do we work out the height of our triangle?
Well, we begin by recalling the formula that helps us calculate the area of a triangle. It’s the length of the base multiplied by its perpendicular height divided by two. You might alternatively have seen this as a half times base times height. Let’s use this formula and fill out what we know.
We’re told that the area of the triangle is 52 square centimetres and that the length of its base is 13. So we can say that 52 must be equal to 13 multiplied by the height all divided by two. Now in fact, let’s use algebra and let ℎ be equal to the height of our triangle. Our job is to work out the height. So we’re going to need to solve this equation for ℎ.
To do so, we’ll begin by multiplying both sides of the equation by two. And so we find that 104 is equal to 13 times ℎ or 13ℎ. Next, we divide through by 13. So ℎ is equal to 104 divided by 13 or 104 over 13. Now in fact, eight thirteens are 104. So 104 divided by 13 must be equal to eight. And so we can say that the height of our triangle is eight centimetres.
And we’re now ready to determine the ratio between its base length and its height. This means that the ratio between the base length and the height of this triangle is 13 to eight. And of course, we must recall that when writing a ratio, order matters. We must put the base length first and then the height. It would not be correct to write eight to 13 as our final answer.