𝐴𝐵𝐶 is a triangle where 𝐴𝐵 equals 50 centimeters, 𝐵𝐶 equals 30 centimeters, and 𝐴𝐶 equals 42 centimeters. Find the area of the triangle, giving the answer to the nearest square centimeter.
To help us understand what’s happening with this question, I’ve just drawn a little sketch of a triangle. Doesn’t necessary represent the scale or the right proportion of our triangle. But what I wanna do is just give us an idea of where the sides are going to be.
So we know that 𝐴𝐵 is equal to 50 centimeters. 𝐵𝐶 is 30 centimeters. And 𝐴𝐶 is 42 centimeters. Okay, so now what we want to actually do is we want to find the area of this triangle. However, we don’t know which is the base, which is the perpendicular height. We don’t know anything like this. So how are we actually going to do it?
Well, to enable us to actually find the area of this triangle, what we’re actually going to use is Heron’s formula. And Heron’s formula tells us that 𝐴, the area of the triangle, is 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 equal to our semiperimeter. So that’s the semiperimeter of our triangle.
And we have a formula for this. And that’s 𝑠 is equal to 𝑎 plus 𝑏 plus 𝑐, because that would be our perimeter of our triangle, and then all divided by two because it’s a semiperimeter.
Okay, great! We’ve got the formula we need. So now let’s go and find the area of our triangle. So first of all, we’re actually going to find 𝑠. And to do that, we’re gonna substitute in our values for 𝑎, 𝑏, and 𝑐. So then when we substitute in our values, we get 30 plus 42 plus 50 all divided by two.
Great! So now we can calculate 𝑠. So 𝑠 is equal to 122 over two, which is equal to 61. Okay, we’ve found our semiperimeter. Now let’s go on and find our area. And to find our area, we’re actually gonna substitute our values into Heron’s formula. So we have 𝐴, our area, is equal to the square root of 61, because that was our 𝑠, that was our semiperimeter. And then this is multiplied by 61 minus 30, cause 30 is our 𝑎, multiplied by 61 minus 42 multiplied by 61 minus 50, which gives us the square root of 61 multiplied by 31 multiplied by 19 multiplied by 11. And then we get a final answer of 628.664.
Okay, not quite finished there. There’s one more thing to do. The question asks for the answer to the nearest square centimeter. So therefore, we can say that the area of the triangle 𝐴𝐵𝐶, where 𝐴𝐵 is equal to 50 centimeters, 𝐵𝐶 is equal to 30 centimeters, and 𝐴𝐶 is equal to 42 centimeters, is equal to 629 centimeters squared to the nearest square centimeter.