Determine, to the nearest hundredth, the area of the trapezoid 𝐴𝐵𝐶𝐷.
So, in this question, what we’re looking to do is find the area of a trapezoid. So to do that, we need a formula. And the formula we have is that the area of a trapezoid is equal to a half 𝑎 plus 𝑏 ℎ. But what are 𝑎, 𝑏, and ℎ?
Well, if you look at these sketches, 𝑎 and 𝑏, well these are the parallel sides and ℎ is the perpendicular height. And as you can see from our sketches, it doesn’t matter about the orientation of the trapezoid; it could be standing up or on its side. Either way, but you need to make sure that the distance between our 𝑎 and 𝑏, this is the perpendicular height.
Well, if you take a look at our diagram, well, we know our 𝑎 and 𝑏. It doesn’t matter which way around we call them. But I’m gonna call the top value 𝑎, that’s 23 centimeters, the bottom value 𝑏, which is 59 centimeters. But then we’ve got ℎ. And we can see on our diagram, we don’t know what this is. So the first thing we need to do is calculate ℎ.
Now, to enable us to do this, there are a couple of methods we could use. And because we have a right triangle, the two methods we can use are trigonometry or Pythagorean theorem. And the method we’re gonna start with is trigonometry. And that’s because for the trigonometry method, you wouldn’t need to find the bottom length, so for the bottom side, because we’ve got one side, an angle, and then this side we want to find.
So the first thing we do is label our sides cause this is what we do with any trigonometry problem like this. So we’ve got the hypotenuse, the opposite, and the adjacent. Then, for the next step, what we need to do is choose which ratio we’re going to use. Well, we’ve got the hypotenuse. And we want to find the adjacent. So if we use SOHCAHTOA, which is just a way to remember our trigonometric ratios, then we can see that the ratio we’re going to use is the cosine ratio. And that’s because it contains the 𝑎 and the ℎ.
So if we substitute in our values, we’re gonna get cos 30 is equal to ℎ over 36. So next, what we do is multiply both sides by 36. So then we have 36 cos 30 is equal to ℎ. Then what we can do at this stage is we can work out what ℎ is by multiplying 36 by cos 30. And if we did that, we get 31.1769 et cetera. However, what the best thing to do is is actually keep in this form cause it saves the accuracy because when we’re carrying out the calculations later, we want to make sure that we don’t lose any accuracy.
Now we did mention that there’s an alternative method. And this method is to use the Pythagorean theorem. But how would we do this? Well, the Pythagorean theorem states that with a right triangle that 𝑐 squared equals 𝑎 squared plus 𝑏 squared, where 𝑐 is the hypotenuse, the longest side of the triangle. Well, we have the hypotenuse, but we don’t have either of the smaller sides.
So what would we do now? Well, we’d have to find out what one of them is so that we could then use them to find ℎ. Well, if we label our right triangle and have 𝑎 as the height and 𝑏 as the length in the bottom, then what we can do is find this length 𝑏.
Now we’re gonna have 𝑏 is equal to a half and then multiplied by 59 minus 23. And that’s because we’ve got an isosceles trapezoid, which means that the overhang, either side on the bottom side, is going to be equal. So therefore, if we do 59 minus 23, then it’s gonna leave us with the overhang, and then divide it by two. And it’d give us the section we’re looking for, which is the bottom of our right triangle or the side 𝑏, which is now gonna be equal to 18.
So now what we can do is use the Pythagorean theorem. And if we do that, what we’re gonna have is 36 squared minus 18 squared equals ℎ squared. And that’s because if we’re using the Pythagorean theorem and want to find a shorter side, we subtract the square of one of the shorter sides away from the square of the hypotenuse. And then this will give us the square of the other shorter side. And if we work this out, we’re gonna get ℎ squared is equal to 972, which means that ℎ is gonna be equal to root 972, which once again will give us 31.1769 et cetera, which is the same as we got with the first method. So, great, but again, what we would do is leave it as root 972 because we’d retain the accuracy.
So now, finally, what we can do is use this value and the other values we have to work out the area of our trapezoid. So therefore, the area is gonna be equal to a half multiplied by, then we’ve got 23 plus 59. And then this is multiplied by 36 cos 30. Or we could’ve had our root 972. Well, this is gonna give us an area of 1278.25349 et cetera.
But have we finished here? Well, no. That’s because the question asks us leave it to nearest hundredth. Well, the nearest hundredth means to two decimal places. So therefore, our area is going to be 1278.25 centimeters squared. And as we said, that’s to two decimal places. And we get that because if we want to round, then if we want to two decimal places or to the nearest hundredth, we look at the five. And then the number to the right of this is our deciding number. And if this is a five or above, we’d round the five to a six. However, if it’s less than five, then what we do is the five remains the same. So we round down. And because it’s a three, we round it down. So our answer was, as we’ve said, 1278.25.