Determine the area of the isosceles triangle 𝐴𝐵𝐶, given that 𝐴𝐵 equals 𝐴𝐶, 𝐴𝐷 is perpendicular to 𝐵𝐶, and 𝐵𝐶 is equal to the square root of 372 plus two centimetres, and 𝐴𝐷 is equal to the square root of 93 minus one centimetres.
First up, let’s write the relevant information onto our diagram. Well, 𝐴𝐵 equals 𝐴𝐶 has already been included for us. 𝐴𝐷 being perpendicular to 𝐵𝐶 has also been included for us. So let’s just mark on these lengths. 𝐵𝐶 is equal to the square root of 372 plus two centimetres, and 𝐴𝐷 is equal to the square root of 93 minus one centimetres.
And also let us recall that the area of a triangle is equal to half times the base times the perpendicular height. So the area of a triangle is a half times the base. Well this is the base, the square root of 372 plus two, times the perpendicular height; well, this is our perpendicular height. So that’s the calculation we’ve got to do. Okay, let’s multiply out the parentheses first.
We’re gonna do root 372 times root 93. Well, that comes out as root 34596, which is just 186. Then, we’ve got root 372 times negative one, so that’s negative root 372. Then, we’ve got positive two times root 93 and positive two times negative one.
Well, before we carry on, let’s look at this: the square root of 372. The 372 can be written as four times 93. And the square root of four times 93 is the same as the square root of four times the square root of 93. And the square root of four is two. So we can write that as two times root 93 or just two root 93.
So now, we’ve got 186 take away two root 93 plus two root 93 take away two. Well, if I start with negative two root 93 and I add two root 93, those two things are gonna cancel each other out and make zero. So in the brackets or in the parentheses, I’ve just got 186 minus two, which is 184. And a half of 184 is 92. And don’t forget that our length units were centimetres, so the area is gonna be in centimetres squared. So the answer is the area is 92 centimetres squared.