Question Video: Finding the Area of a Quadrilateral given the Side Lengths and the Measure of an Angle Mathematics

Video Transcript

Find the area of the quadrilateral giving the answer to three decimal places.

But what I’m gonna do first to actually solve this problem is divide our quadrilateral into two triangles. So we’ve got triangle 𝐴, which is a right triangle, and triangle 𝐵. Now, before we can actually work out the area of either triangle, what we need to find is the length 𝐵𝐷. And what we’re gonna do to help us find the length of 𝐵𝐷 is use the Pythagorean theorem. And we can use the Pythagorean theorem because we’ve got a right triangle. And we know that because we can see that there’s a right angle here at 𝐴.

And the Pythagorean theorem tells us that if we have a right triangle with the sides 𝐴, 𝐵, and 𝐶, where 𝐶 is the hypotenuse, so it’s the longest side opposite the right angle, then 𝐴 squared plus 𝐵 squared is gonna be equal to 𝐶 squared. Well, if we take a look at our diagram, we can see that 𝐵𝐷 is actually a hypotenuse because it’s opposite our right angle at 𝐴. So therefore, we can say that 𝐵𝐷 squared equals 18 squared plus 24 squared. So therefore, we get 𝐵𝐷 squared is equal to 900. Then if we square root both sides, we get 𝐵𝐷 is equal to 30 meters. So fantastic, we’ve actually found the missing side.

Okay, so now we’ve actually found the missing side. What we can do is move straight on and start to find the area of each of our triangles. So I’m going to start with triangle 𝐴. Well, for triangle 𝐴, we know that the area is gonna be equal to half the base times the height. And that’s because we’ve actually got a right triangle. So therefore, we know the perpendicular height. So this is gonna be equal to a half multiplied by 24 multiplied by 18 which is gonna be equal to 216 meters squared. So fantastic, we’ve actually dealt with triangle 𝐴.

So now, let’s move on to triangle 𝐵. Well, for triangle 𝐵, it’s not quite as straightforward because we actually don’t know the perpendicular height. So therefore, what we’re gonna use is Heron’s formula to help us find the area of this triangle. And Heron’s formula tells us that if we have triangle 𝐴𝐵𝐶, then the area is actually equal to the square root of 𝑠 multiplied by 𝑠 minus 𝐴 multiplied by 𝑠 minus 𝐵 multiplied by 𝑠 minus 𝐶, where 𝑠 is actually the semiperimeter which can be found by finding the perimeter of our triangle — so adding 𝐴, 𝐵, and 𝐶 — and then dividing it by two.

Okay, so now we’ve got Heron’s formula. And we know what 𝑠 is. Let’s use these to actually find the area of triangle 𝐵. So first, we’re going to find our semiperimeter. And this is gonna be equal to 15 plus 30 plus 37 over two which is equal to 41 because 15 plus 30 plus 37 gives us 82. And 82 over two gives us 41. Okay, now we’ve got this. We can actually use Heron’s formula to find our area. So we can say that the area is equal to 41 multiplied by 41 minus 15 multiplied by 41 minus 30 multiplied by 41 minus 37. So therefore, this is equal to the square root of 46904 meters squared.

Now, we’ll leave it actually as a square root. And the reason I’m doing that is for accuracy cause we’re now gonna bring them together and find the area of the total quadrilateral. And I don’t want to lose any accuracy at this point. So now, we’re actually gonna find the total area of the quadrilateral. And to do that, we’re gonna add the area of triangle 𝐴 and the area of triangle 𝐵. So we’ve got 216 plus the root of 46904. Well, this gives us 432.5733132.

So it’s at this point we look back at the question to check how we should leave our answer. We can see that it wants it to three decimal places. So therefore, we can say that the total area of the quadrilateral is equal to 432.573 meters squared, to three decimal places.