What is the area of the triangle in square units? (1 point) Responses 6 6 15 15 18 18 9

Triangle corner a: (0, 3)
Triangle corner b (10, 6)
triangle corner c: (6, 3)

1 answer

To find the area of a triangle given the coordinates of its vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\), we can use the formula:

\[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]

Let's assign the coordinates of the triangle corners:

  • \(A (x_1, y_1) = (0, 3)\)
  • \(B (x_2, y_2) = (10, 6)\)
  • \(C (x_3, y_3) = (6, 3)\)

Substituting these coordinates into the formula:

\[ \text{Area} = \frac{1}{2} \left| 0(6 - 3) + 10(3 - 3) + 6(3 - 6) \right| \]

Let's calculate each term inside the absolute value:

  1. \(0(6 - 3) = 0\)
  2. \(10(3 - 3) = 0\)
  3. \(6(3 - 6) = 6 \times (-3) = -18\)

Now substituting these back into the formula:

\[ \text{Area} = \frac{1}{2} \left| 0 + 0 - 18 \right| = \frac{1}{2} \left| -18 \right| = \frac{1}{2} \times 18 = 9 \]

Thus, the area of the triangle is:

\[ \boxed{9} \text{ square units} \]