https://file.aiquickdraw.com/mathbot/file/1725647280968bb9bv31o.jpg

In the image provided, we see a coordinate plane with three points labeled as **Classroom 1**, **Classroom 2**, and **Classroom 3**. These three points form a triangle, and the problem asks to find the area of the triangle (garden).

Here’s the step-by-step solution to finding the area of a triangle given its vertices on the coordinate plane.

The coordinates of the three classrooms are approximately:
- **Classroom 1**: (−3, 2)
- **Classroom 2**: (2, 5)
- **Classroom 3**: (4, −3)

We can use the **shoelace formula** (also known as the **determinant formula**) to calculate the area of the triangle given its vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\):

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

Substitute the coordinates of the three classrooms:
\[
\text{Area} = \frac{1}{2} \left| (-3)(5 - (-3)) + 2((-3) - 2) + 4(2 - 5) \right|
\]
\[
= \frac{1}{2} \left| (-3)(8) + 2(-5) + 4(-3) \right|
\]
\[
= \frac{1}{2} \left| -24 - 10 - 12 \right|
\]
\[
= \frac{1}{2} \left| -46 \right| = \frac{1}{2} \times 46 = 23
\]

So, the area of the triangle (garden) is **23 square units**.

Would you like a more detailed explanation or further clarification? 

Here are five related questions:
1. How would the area change if Classroom 3 were moved to a different point?
2. What is the significance of using the shoelace formula in this context?
3. Can you calculate the perimeter of the triangle formed by the three classrooms?
4. How would you approach finding the centroid of the triangle?
5. What would be the area if the classrooms were placed on a different set of coordinates?

**Tip:** The shoelace formula is very useful for finding areas of polygons when their vertices are known on a coordinate plane.

A middle school has three modular classrooms outside. The graph shows where to place a garden in the area between the three modular classrooms. What is the area of the garden?

Learn how to calculate the area of a garden formed by three classrooms on a coordinate plane using the shoelace formula. This problem involves key geometry concepts, including coordinate geometry and the shoelace theorem, suitable for students in Grades 8-10.

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