Given the figure shown, determine the area of a similar polygon 𝐴 prime 𝐵 prime 𝐶 prime in which 𝐴 prime 𝐵 prime equals three.
Well, the first thing we need to look at is where it says that we’re looking at a similar polygon. But what does this mean? Well, a similar polygon means that if two polygons are, in fact, similar to each other, then corresponding sides are proportional and their angles are equal. Another way we can think about it is that, in fact, one is an enlargement of the other. So, in that case, what we can do to help us solve this problem is find out what our scale factor of enlargement would be.
And what we have is a formula for scale factor. And that is that the scale factor is equal to the new length over the original length. Well, as in our new polygon 𝐴 prime 𝐵 prime 𝐶 prime, we’re looking at the line 𝐴 prime 𝐵 prime and we know that it’s three, then the corresponding line in our triangle that we have here on the grid is 𝐴𝐵. And we can see that the length of this corresponding line is five units. So, therefore, our scale factor is gonna be our new length, which is three, divided by the original length, which is five. So, we can say the scale factor is three-fifths.
Well, in the question, what we’re asked to do is find the area of the new triangle or the new polygon. Well, we know that the area of a triangle is equal to a half the base times the height. And that’s where the height is the perpendicular height. Well, if we take a look at the sketch that we’ve drawn, we know the base because that’s three. But what we don’t know is the perpendicular height. So how are we going to work this out?
Well, what we can do is utilize the scale factor that we found. And that’s because if we take a look at the original triangle, then what we have is a height of seven units. So, therefore, if we want to find out what ℎ is, so the height, on our new triangle, then it’s gonna be equal to seven multiplied by 0.6. And it’s 0.6 because the scale factor we found to be three-fifths or three over five, and this is the same as 0.6 when put into a decimal. So, therefore, our height is gonna be equal to 4.2.
So, now, what we can do is use the formula for the area of a triangle. And what we’re gonna do is multiply a half by the base by the height. So, we’re gonna have a half multiplied by three multiplied by 4.2. Well, this’s gonna give us 6.3. So, we can say that given the figure shown, the area of a similar polygon 𝐴 prime 𝐵 prime 𝐶 prime in which 𝐴 prime 𝐵 prime equals three is 6.3.