How do you find the area of the triangle with the given vertices A(2, -4), B(1,3), C(-2, -1)?

Question

Sadie Allen · Accepted Answer

12.5 sq units

Area of a triangle with vertices #(x_1, y_1), (x_2, y_2), (x_3,y_3)# is #1/2[x_1(y_2-y_3) +x_2 (y_3-y_1) +x_3(y_1 -y_2)]# Using this formula the required area would be #1/2[2(3+1)+1(-1+4)+(-2)(-4-3)]# #1/2 [8+3+14]= 25/2 =12.5#

Avery Allen · Answer

undefined To find the area of the triangle with the given vertices A(2, -4), B(1,3), C(-2, -1), you can use the formula for the area of a triangle formed by three points in the Cartesian plane. Here are the steps:

1. Use the coordinates of the given vertices to find the lengths of the three sides of the triangle using the distance formula.

2. Once you have the lengths of the three sides (denote them as $a$, $b$, and $c$), you can use Heron's formula to calculate the area of the triangle.

3. Heron's formula states that the area ($A$) of a triangle with side lengths $a$, $b$, and $c$ is given by:
   $$ A = \sqrt{s(s-a)(s-b)(s-c)} $$
   where $s$ is the semi-perimeter of the triangle, calculated as:
   $$ s = \frac{a + b + c}{2} $$

4. Substitute the lengths of the sides into the formula and calculate the area.

Following these steps will give you the area of the triangle formed by the given vertices.