
(5,0),(−2,6)
 and (−2,0)
Step 2 of 2 :  What is the area of the triangle? (Hint: Make use of the midpoint formula if the triangle is isosceles.)

To find the area of the triangle with vertices at \((5, 0)\), \((-2, 6)\), and \((-2, 0)\), we can follow these steps:

### Step 1: Check if the Triangle is Isosceles
A triangle is isosceles if two of its sides are of equal length. We calculate the lengths of the sides using the distance formula:

\[
\text{Distance between } (x_1, y_1) \text{ and } (x_2, y_2) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

1. **Length of side between \((5, 0)\) and \((-2, 6)\):**
\[
d_1 = \sqrt{((-2) - 5)^2 + (6 - 0)^2} = \sqrt{(-7)^2 + 6^2} = \sqrt{49 + 36} = \sqrt{85}
\]

2. **Length of side between \((5, 0)\) and \((-2, 0)\):**
\[
d_2 = \sqrt{((-2) - 5)^2 + (0 - 0)^2} = \sqrt{(-7)^2} = 7
\]

3. **Length of side between \((-2, 6)\) and \((-2, 0)\):**
\[
d_3 = \sqrt{((-2) - (-2))^2 + (6 - 0)^2} = \sqrt{0 + 6^2} = 6
\]

The sides \(d_2\) and \(d_3\) are not equal to \(d_1\), and they aren't equal to each other either. So, the triangle is not isosceles, and we don't need the midpoint formula.

### Step 2: Use the Area Formula for a Triangle
For a general triangle with vertices \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\), the area \(A\) can be calculated using the following formula:

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

Substitute the coordinates of the points into the formula:

\[
A = \frac{1}{2} \left| 5(6 - 0) + (-2)(0 - 0) + (-2)(0 - 6) \right|
\]

\[
A = \frac{1}{2} \left| 5(6) + 0 + (-2)(-6) \right|
\]

\[
A = \frac{1}{2} \left| 30 + 12 \right| = \frac{1}{2} \times 42 = 21
\]

### Final Answer:
The area of the triangle is **21 square units**.

---

Would you like more details on any part of the solution, or do you have any further questions?

### Related Questions:
1. How can we determine if a triangle is isosceles using only the coordinates?
2. What is the formula to calculate the perimeter of a triangle given its vertices?
3. How do you find the centroid of a triangle using its vertices?
4. What is the significance of the determinant in calculating the area of a triangle?
5. How would the area formula change if the triangle were in 3D space?

### Tip:
When dealing with coordinate geometry problems, drawing a rough sketch of the points on the coordinate plane can often help you visualize the problem better and avoid mistakes.

Learn how to calculate the area of a triangle with vertices at (5, 0), (-2, 6), and (-2, 0) using coordinate geometry. This problem involves applying the distance formula and the area of triangle formula. Suitable for high school students.

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