Question Video: Finding the Measure of the Smallest Angle in a Triangle given the Ratio between Its Three Angles

The ratio of the three angles in a triangle is 5 : 4 : 9. Find the size of the smallest angle.

03:08

Video Transcript

The ratio of the three angles in a triangle is five to four to nine. Find the size of the smallest angle.

So, with questions involving ratio, there are lots of different approaches that you can take. I’ll demonstrate two of these approaches, and then you could decide which of them you prefer. So, the first approach is an algebraic approach, where we say, well, we don’t know what these angles are, but we know they’re in the ratio of five to four to nine. Which means I could call these angles five 𝑥, four 𝑥, and nine 𝑥, where 𝑥 is just representing some unknown value. But this keeps the five to four to nine ratio.

Now, remember our key fact, that the sum of the angles in a triangle is 180 degrees. So, I can turn this into an equation. So, if I add plus signs between those three terms, then it’s equal to 180. So, what I’ve done is set up an equation involving this unknown letter 𝑥. So, now, I can simplify this equation. Add five 𝑥 plus four 𝑥 plus nine 𝑥 becomes 18 𝑥. So, I have 18 𝑥 is equal to 180. To solve the equation then, I need to divide both sides of the equation by 18, and this gives me that 𝑥 is equal to 10.

Now, the question, remember, said find the size of the smallest angle. So, the smallest angle is this four 𝑥, the one that has the least parts of this ratio. So, in order to work out the smallest angle, I need to multiply 𝑥 by four. So, I have four 𝑥. Four times 10 is equal to forty. And this gives me the answer to the problem, which is that the size of the smallest angle is 40 degrees.

So, that’s one way of approaching it, treating it like an algebra problem and setting up an equation. The other way that I’d like to think about ratio problems, is thinking about the parts of the ratio. So, we have a ratio of five to four to nine. If I add them together, five plus four plus nine is 18. So, in total, there are 18 equal parts in this ratio. Now, as we’ve said many times in this video, the sum of the angles in a triangle is 180 degrees. So, those 18 parts together are worth 180. I want to work out the size of the smallest angle, so I want to know what four parts are worth.

And there are lots of different ways I could do this. I could work out what one part is, by dividing by 18. So, that would give me one part is equal to 10. And then, to find four parts, I’d have to multiply by four. And so, of course, that gives me the same answer, as before, of forty degrees. I could, perhaps, have approached this ratio in a slightly different way.

Instead of finding one part, I could’ve found, perhaps, two parts. So, that would’ve meant dividing both sides by nine, and then I’d just have doubled it to find the four parts as 40. So, whichever approach you prefer, either an algebraic approach involving setting up an equation or thinking about ratio in terms of equal parts and dividing down and then scaling back up to however many parts you’re looking for.

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