Question

Find the area of triangle ABC whose vertices are A (-5, 7), B (-4, -5), and C (4, 5).

a) 48 square units
b) 72 square units
c) 36 square units
d) 24 square units

Answer 1

Upon calculating the determinant, we find the area of triangle ABC with the coordinates given to be 53 square units. However, this correct area is not listed among the provided options. There may be a mistake in the question or given options.

Step-by-step explanation:

To find the area of triangle ABC whose vertices are A (-5, 7), B (-4, -5), and C (4, 5), we can use the determinant method derived from the Shoelace formula for the coordinates of the vertices. Here is the step-by-step process:

1. Arrange the coordinates in a grid to align like vertices:
   
   
   A (-5, 7)
   B (-4, -5)
   C (4, 5)
   A (-5, 7) (Repeat the first vertex at the end for calculation ease)
2. Calculate the determinant using the diagonals method (also known as the Shoelace formula):
   
   
   (First diagonal) = (-5) * (-5) + (-4) * (5) + (4) * (7) = 25 + (-20) + 28 = 33
   (Second diagonal) = (7) * (-4) + (-5) * (4) + (5) * (-5) = (-28) + (-20) + (-25) = -73
   
3. Subtract the sum of the second diagonal from the sum of the first diagonal:
   
   
   Absolute difference = |33 + 73| = 106
   
4. Divide the absolute difference by 2 to get the area of the triangle:
   
   
   Area = 1/2 * 106 = 53 square units

None of the given options (48, 72, 36, 24 square units) match the calculated area, which is 53 square units. Therefore, there may be an error in the question or the given options, as the correct area does not appear to be listed.