In the given figure, 𝐻𝐴𝐷 is an equilateral triangle with a perimeter of 45 centimeters. Given that the ratio 𝐴𝐷 to 𝐴𝐵 is equal to the ratio three to seven, determine the area of the rectangle 𝐴𝐵𝐶𝐷.
So, the first bits of information we’re gonna use is that the triangle 𝐻𝐴𝐷 is an equilateral triangle and that its perimeter is 45 centimeters. Well, what we know about an equilateral triangle is that, in fact, it has all angles the same, so that’s all interior angles are the same, and all side lengths are the same. So, what I’ve done here is labeled the sides 𝑥. So, therefore, we know that three 𝑥 is gonna be equal to 45 because the perimeter is the distance around the outside.
And if we add 𝑥, 𝑥, and 𝑥, we get three 𝑥. And if we divide each side of the equation by three, we get 𝑥 is equal to 15 centimeters. So, therefore, we now know that each of the sides of our triangle are 15 centimeters. But we also know that the length 𝐴𝐷, which is one of the side lengths of our rectangle, is also 15 centimeters.
Well, now, the next bit of information we’re gonna use is that the ratio 𝐴𝐷 to 𝐴𝐵 is equal to the ratio three to seven. Well, now, we actually know the length of 𝐴𝐷 because that’s 15 centimeters, but we don’t know the length of 𝐴𝐵. But we can work this out because what we do is take a look at our ratio. And we can see that to get from three to 15, because we know that the side length 𝐴𝐷 is 15 centimeters long, we have to multiply by five. So, therefore, whatever we’ve done to one side of the ratio, we must do to the other. So, we multiply the other side by five. So, it’s seven by five, and it gives us 35 centimeters. So, we now know that the length 𝐴𝐵 is 35 centimeters.
So, great, have we solved the problem? Well, not quite. And that’s because the question asked us to determine the area of the rectangle. So, we know that the area of a rectangle is equal to the length times the width. So, the area of the rectangle 𝐴𝐵𝐶𝐷 is gonna be equal to 35 multiplied by 15, which will give us a final area of 525, and then the units are centimeters squared.
Now, we could have found this using a calculator. But a quick mental method we could also use was the one shown here. Do 10 multiplied by 35 is 350. Then, half of this is what five multiplied by 35 would be, which would be 175. Add them together, gives us 525.