27 Triangle $$ M A X $$ has vertices with coordinates $$ M(-5,-2), A(1,4) $$, and $$ X(4,1) $$. Determine and state the area of $$ riangle M A X $$.

Question

27 Triangle $$ M A X $$ has vertices with coordinates $$ M(-5,-2), A(1,4) $$, and $$ X(4,1) $$. Determine and state the area of $$ 	riangle M A X $$.

UpStudy AI · Accepted Answer

To find the area of triangle $$ \triangle MAX $$ with vertices $$ M(-5, -2) $$, $$ A(1, 4) $$, and $$ X(4, 1) $$, we can use the **shoelace formula**. The shoelace formula for the area of a triangle with vertices $$(x_1, y_1)$$, $$(x_2, y_2)$$, and $$(x_3, y_3)$$ is:

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

Let's apply this step-by-step:

1. **Identify the coordinates:**
   - $$ M(-5, -2) $$: $$ x_1 = -5 $$, $$ y_1 = -2 $$
   - $$ A(1, 4) $$: $$ x_2 = 1 $$, $$ y_2 = 4 $$
   - $$ X(4, 1) $$: $$ x_3 = 4 $$, $$ y_3 = 1 $$

2. **Plug the values into the formula:**
   $$
   \text{Area} = \frac{1}{2} \left| (-5)(4 - 1) + 1(1 - (-2)) + 4(-2 - 4) \right|
   $$

3. **Calculate each term:**
   $$
   (-5)(3) = -15
   $$
   $$
   1(3) = 3
   $$
   $$
   4(-6) = -24
   $$

4. **Sum the terms:**
   $$
   -15 + 3 - 24 = -36
   $$

5. **Take the absolute value and divide by 2:**
   $$
   \text{Area} = \frac{1}{2} \times 36 = 18
   $$

**Therefore, the area of $$ \triangle MAX $$ is 18 square units.**

**Answer:**  
The area of triangle $$ \triangle MAX $$ is 18 square units.
The area of triangle $$ \triangle MAX $$ is 18 square units.

27 Triangle \( M A X \) has vertices with coordinates \( M(-5,-2), A(1,4) \), and \( X(4,1) \). Determine and state the area of \( \triangle M A X \).

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