Question
What is the Area of this Triangle? (A 0,3 B 10,6 C 6,3) Responses 15 units2 15 units2 6 units2 6 units2 18 units2 18 units2 9 units2
Answers
GPT-4o mini
To find the area of triangle ABC with vertices A(0, 3), B(10, 6), and C(6, 3), we can use the formula for the area of a triangle given its vertices at points \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\):
\[
\text{Area} = \frac{1}{2} \left| x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) \right|
\]
Assigning the coordinates:
- \(A(0, 3) \rightarrow (x_1, y_1) = (0, 3)\)
- \(B(10, 6) \rightarrow (x_2, y_2) = (10, 6)\)
- \(C(6, 3) \rightarrow (x_3, y_3) = (6, 3)\)
Substituting into the formula, we have:
\[
\text{Area} = \frac{1}{2} \left| 0(6-3) + 10(3-3) + 6(3-6) \right|
\]
This simplifies to:
\[
\text{Area} = \frac{1}{2} \left| 0 + 0 + 6(-3) \right| = \frac{1}{2} \left| -18 \right| = \frac{1}{2} \times 18 = 9
\]
Thus, the area of triangle ABC is \(9\) square units.
Therefore, the correct response is **9 units²**.
\[
\text{Area} = \frac{1}{2} \left| x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) \right|
\]
Assigning the coordinates:
- \(A(0, 3) \rightarrow (x_1, y_1) = (0, 3)\)
- \(B(10, 6) \rightarrow (x_2, y_2) = (10, 6)\)
- \(C(6, 3) \rightarrow (x_3, y_3) = (6, 3)\)
Substituting into the formula, we have:
\[
\text{Area} = \frac{1}{2} \left| 0(6-3) + 10(3-3) + 6(3-6) \right|
\]
This simplifies to:
\[
\text{Area} = \frac{1}{2} \left| 0 + 0 + 6(-3) \right| = \frac{1}{2} \left| -18 \right| = \frac{1}{2} \times 18 = 9
\]
Thus, the area of triangle ABC is \(9\) square units.
Therefore, the correct response is **9 units²**.