Find the area of the triangle below using determinants.
Here, we’ve labelled the vertices. Before we find the area using determinants, there is a different way to solve this problem. The area of a triangle is equal to one-half times the base of the triangle times the height of the triangle. So if we would let this be our base, which is four, and then the height is exactly perpendicular has to go straight down or straight up, if it was upwards, from the base, so the height of this triangle would be nine. So we have one-half times four times nine. And one-half of four is two. So two times nine means our area should be 18.
Now, this is not how we were supposed to solve it we’re supposed to solve it using determinants, which we will do. But it’s always a good idea if you know another way to solve a problem; maybe just do it just to double check. To find the area of a triangle using determinants, area is equal to one-half times the determinate of this three-by-three matrix.
So what are 𝘢, 𝘣, 𝘤, 𝘥, 𝘦, and 𝘧? Well, the vertices of the triangle are 𝑥, which is the point 𝘢, 𝘣, 𝑦, which is the point 𝘤, 𝘥, and 𝑧, which is the point 𝘦, 𝘧. So we can go ahead and just choose whichever points we want to be 𝑥, 𝑦, and 𝑧. So 𝑥 is zero, five, 𝑦 is four, five, and 𝑧 is three, negative four. So we can replace 𝘢 and 𝘣 with zero, five; 𝘤, 𝘥 with four, five; and 𝘦, 𝘧 with three, negative four.
So now how do we evaluate the determinant of this three-by-three matrix? Let’s first write down the one-half, so we don’t forget to multiply by at the end. So we take zero times the determinant of the numbers that are not in the row or column with the zero. Then we subtract five times the determinant of the numbers that are not in the row or column of the five. And then we add one times the determinant of the numbers that are not in the row or column of the one.
So how do we find the determinant of a matrix? We take 𝘢 times 𝘥 minus 𝘣 times 𝘤; it’s almost like cross multiplying and subtracting them. So bring down the one-half. Now before we evaluate that first determinant in the pink, it’s zero times that determinant and zero times anything will be zero. So there’s no point of wasting our time evaluating that because we’ll get zero for that anyway. So let’s begin with negative five times that determinant.
So we’ve negative five and then we start to evaluate the determinant. Four times one minus one times three plus one times four times negative four minus five times three. After multiplying the numbers on the innermost parentheses, which are brackets, now we need to subtract these numbers. Next, we need to multiply. So we’ve negative five minus 31. And now, we need to subtractive those. So we have one-half times negative 36, which is negative 18. However, this is an area, and an area is a measurement which needs to be positive. So the area of this triangle will be 18 units squared.