Question Video: Finding the Base Length of a Triangle given Its Area and the Equality of Base and Height | Nagwa Question Video: Finding the Base Length of a Triangle given Its Area and the Equality of Base and Height | Nagwa

Question Video: Finding the Base Length of a Triangle given Its Area and the Equality of Base and Height Mathematics • Second Year of Preparatory School

In a triangle whose base and height are equal, the base length is given by 𝑏 = √2𝐴, where 𝐴 is the area of the triangle. Estimate, to the nearest whole number, the base length of this triangle if its area is 17 square metres.

03:36

Video Transcript

In a triangle whose base and height are equal, the base length is given by 𝑏 equals root two 𝐴, where 𝐴 is the area of the triangle. Estimate, to the nearest whole number, the base length of this triangle if its area is 17 square metres.

So, to help us understand what’s going on and picture the question, what I’ve done is a little sketch where the base and height are equal. And I’ve called both of these 𝑏. And the area 𝐴 is 17 square metres. And the other bit of information we’re given from the question is that 𝑏 is equal to root two 𝐴. In order to solve this problem, there are two methods. So, I’m going to show you both.

So, first of all, we have a formula to help us calculate the area of a triangle. And that formula is that the area is equal to the half the base times the height. So then, if we use the values we’ve got, we’ve got 17, because that’s our area, is equal to a half multiplied by 𝑏 squared. And that’s because it’s 𝑏 multiplied by 𝑏, because our height and our base are the same. So, it’s gonna be 𝑏 squared.

So then, to solve this equation to help us find 𝑏, what we’re gonna do first is multiply each side of the equation by two. That’s cause we’ve got a half 𝑏 squared, a half multiplied by 𝑏 squared. So therefore, if we multiply by two, we’ll have a whole 𝑏 squared. So therefore, when we multiply each side of the equation by two, we’re gonna get 34, because 17 multiplied by two is 34, is equal to 𝑏 squared. So, next what we’ll do is we take the square root of both sides of the equation. And when we do that, we get root 34 is equal to 𝑏. What does that mean that we found the value of 𝑏? Have we solved the problem?

Well, no. We’re asked to estimate to the nearest whole number the base length of the triangle. So, we need to do that now using our root 34. And we’re gonna do that in a moment, once we’ve used the other method to show us how to get to the length of 𝑏 as well.

For the second method, we’re told that 𝑏 is equal to root two 𝐴. So therefore, 𝑏 is gonna be equal to the square root of two multiplied by 17. That’s because 17 is our 𝐴. So therefore, this gives us root 34, because two multiplied by 17 is 34, which is the same answer we got from the first method. So, now what we’re going to do is work out what this is, estimated to the nearest whole number, as I said we’d come back to.

Well, if we’re consider root 34, we can first think, are there two square numbers that 34 is between? Well, yes, there are. Because 34 is between 25 and 36. And we’re looking at square numbers, because we’re dealing with square roots here. And we can say that root 34, therefore, must be greater than root 25 but less than root 36. So therefore, we can say that root 34 must be greater than five, because the square root of 25 is five, and less than six, because the square root of 36 is six.

Well, we now need to decide which one it’s gonna be closest to, five or six? Well, if we think about root 34, root 34 is almost equal to root 36, cause 34 is almost equal to 36. So, it’s gonna be closer to root 36 than it will be to root 25. And therefore, what we can say is that an estimate for the base length is six metres.

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