https://file.aiquickdraw.com/mathbot/file/17254772016118g1b848w.jpg

The problem in the image involves a triangle \( ABC \) with vertices at points \( A(0, 0) \), \( B(2, 3) \), and \( C(3, 0) \). The task is to use the determinant method to find the area of triangle \( ABC \).

### Using the Determinant Formula
The area \( \text{Area} \) of a triangle given its vertices \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) can be found using the formula:

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

### Substituting the Coordinates
For the triangle \( ABC \) with vertices \( A(0, 0) \), \( B(2, 3) \), and \( C(3, 0) \), the coordinates are:
- \( (x_1, y_1) = (0, 0) \)
- \( (x_2, y_2) = (2, 3) \)
- \( (x_3, y_3) = (3, 0) \)

Substituting these into the formula:

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

Simplifying the expression:

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

\[
\text{Area} = \frac{1}{2} \left| -9 \right| = \frac{1}{2} \times 9 = 4.5
\]

### Final Answer
The area of triangle \( ABC \) is \( 4.5 \) square units.

---

Would you like more details or have any questions on this method? Here are some related questions you might find interesting:

1. How does the determinant method relate to the geometric interpretation of the area of a triangle?
2. What is the importance of absolute value in the determinant formula for area?
3. Can this method be applied to find the area of a polygon with more than three sides?
4. How would the area change if one of the vertices were moved to a different coordinate?
5. What are some alternative methods to find the area of a triangle given its vertices?

**Tip:** The determinant method is particularly useful for finding the area when the vertices are not aligned with the coordinate axes, as it directly uses the coordinates to compute the area.

Find the area of triangle ABC with vertices A(0, 0), B(2, 3), and C(3, 0) using the determinant method.

Learn how to find the area of triangle ABC with vertices at A(0,0), B(2,3), and C(3,0) using the determinant method. This problem involves key concepts in geometry and coordinate geometry, perfect for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation