Hello, I have another question from my linear algebra class. I'm asked to find the area of a triangle using determinants, but they don't give me the vertices, only the sides. Where A,B,C and D are the vertices, u and v are 2 out of the 3 sides. Now they give me u=(-3,1) and v=(0,2). My question is, can I still use the formula 0.5[x2-x1 + y2-y1] even if the coordinates I have are not those of the vertices?
Thank you
sorry I wrote D, but there is no D, just AB and C
Hello! In order to find the area of a triangle using determinants, you will indeed need the coordinates of the vertices. However, with the side lengths you have provided, you can use a different approach to find the area.
To calculate the area of a triangle using the side lengths u and v, you can use Heron's formula. Heron's formula states that the area of a triangle with side lengths a, b, and c is given by:
Area = √(s(s-a)(s-b)(s-c))
where s is the semi-perimeter of the triangle, calculated as:
s = (a + b + c)/2
In this case, you have the side lengths u and v, so you can substitute them into the formula. Let's calculate step by step:
1. Compute the lengths of the sides:
u = (-3, 1) => |u| = √((-3)^2 + 1^2) = √(10)
v = (0, 2) => |v| = √(0^2 + 2^2) = 2
So, u = √(10) and v = 2.
2. Calculate the semi-perimeter:
s = (u + v + w)/2 = (√(10) + 2 + w)/2
At this point, we need to find the length of the remaining side, w. However, since we only have u and v as side lengths, we cannot directly calculate w.
Therefore, without the coordinates of the vertices, it is not possible to apply Heron's formula or any other methods that require the knowledge of vertex coordinates. You will need the coordinates of the vertices to proceed with finding the area of the triangle using determinants.