Question

What is the area of a triangle whose vertices are R(-4, 2), S(1, 2), and T(-5, -4)?

Answer 1

The area of the triangle with vertices at R(-4, 2), S(1, 2), and T(-5, -4) is calculated to be 17 square units by setting up a matrix with the vertices' coordinates, computing its determinant, and taking the absolute value of half of the determinant result.

Step-by-step explanation:

To calculate the area of a triangle with vertices given by coordinates, one method we can use is the determinant formula. Given the vertices R(-4, 2), S(1, 2), and T(-5, -4), we can use the following steps:

1. Set up the matrix using the coordinates of the vertices:
   
   | -4 2 1 |
   
   | 1 2 1 |
   
   | -5 -4 1 |
2. Compute the determinant of the matrix:
   
   | -4 2 1 |
   
   | 1 2 1 | = -4(2*(-4) -1*1) - 2(1*(-4) -1*(-5)) + 1(1*2 - 2*1)
   
   | -5 -4 1 |
3. Calculate the absolute value of half of the determinant to find the area:
   
   Area = ½ | -4*-8 - 2*-1 + 1*0 | = ½ | 32 + 2 | = ½ * 34 = 17 square units.

The area of the triangle with the given vertices is therefore 17 square units.