Points `(8, 2), (-2, -2)` and (3, 0) are the vertices of

Options

A. an equilateral triangle
B. an isosceles triangle
C. right angled triangle
D. none of these

Answer

To determine the type of triangle formed by the points \( A(8, 2) \), \( B(-2, -2) \), and \( C(3, 0) \), we will calculate the lengths of the sides of the triangle formed by these points.

### Step 1: Calculate the length of side AB
The length of side \( AB \) can be calculated using the distance formula:
\[ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates of points \( A(8, 2) \) and \( B(-2, -2) \):
\[ AB = \sqrt{((-2) - 8)^2 + ((-2) - 2)^2} \]
\[ = \sqrt{(-10)^2 + (-4)^2} \]
\[ = \sqrt{100 + 16} = \sqrt{116} = 2\sqrt{29} \]

### Step 2: Calculate the length of side AC
Now, we calculate the length of side \( AC \):
\[ AC = \sqrt{(3 - 8)^2 + (0 - 2)^2} \]
\[ = \sqrt{(-5)^2 + (-2)^2} \]
\[ = \sqrt{25 + 4} = \sqrt{29} \]

### Step 3: Calculate the length of side BC
Next, we calculate the length of side \( BC \):
\[ BC = \sqrt{(3 - (-2))^2 + (0 - (-2))^2} \]
\[ = \sqrt{(3 + 2)^2 + (0 + 2)^2} \]
\[ = \sqrt{5^2 + 2^2} \]
\[ = \sqrt{25 + 4} = \sqrt{29} \]

### Step 4: Compare the lengths of the sides
Now we have:
- \( AB = 2\sqrt{29} \)
- \( AC = \sqrt{29} \)
- \( BC = \sqrt{29} \)

### Step 5: Check for collinearity
To check if the points are collinear, we can use the property that if the sum of the lengths of two sides equals the length of the third side, then the points are collinear:
\[ AB = AC + BC \]
Substituting the values:
\[ 2\sqrt{29} = \sqrt{29} + \sqrt{29} \]
This equation holds true, indicating that points \( A \), \( B \), and \( C \) are collinear.

### Conclusion
Since the points are collinear, they do not form a triangle.

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