Whether the points (0,7,10),(1,6,-6) and (4,9,-6) from an isosceles triangle

Answer

To determine whether the points \( A(0, 7, 10) \), \( B(1, 6, -6) \), and \( C(4, 9, -6) \) form an isosceles triangle, we need to calculate the lengths of the sides of the triangle formed by these points and check if any two sides are equal.

### Step 1: Calculate the Length of Side AB
Using the distance formula in three-dimensional space, the length \( AB \) is given by:
\[ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]
Substituting the coordinates of points \( A \) and \( B \):
\[ AB = \sqrt{(1 - 0)^2 + (6 - 7)^2 + (-6 - 10)^2} \]
Calculating each term:
\[ AB = \sqrt{(1)^2 + (-1)^2 + (-16)^2} = \sqrt{1 + 1 + 256} = \sqrt{258} \]

### Step 2: Calculate the Length of Side BC
Now, we calculate the length \( BC \):
\[ BC = \sqrt{(4 - 1)^2 + (9 - 6)^2 + (-6 - (-6))^2} \]
Calculating each term:
\[ BC = \sqrt{(3)^2 + (3)^2 + (0)^2} = \sqrt{9 + 9 + 0} = \sqrt{18} \]

### Step 3: Calculate the Length of Side AC
Next, we calculate the length \( AC \):
\[ AC = \sqrt{(4 - 0)^2 + (9 - 7)^2 + (-6 - 10)^2} \]
Calculating each term:
\[ AC = \sqrt{(4)^2 + (2)^2 + (-16)^2} = \sqrt{16 + 4 + 256} = \sqrt{276} \]

### Step 4: Compare the Lengths
Now we have the lengths of all three sides:
- \( AB = \sqrt{258} \)
- \( BC = \sqrt{18} \)
- \( AC = \sqrt{276} \)

To determine if the triangle is isosceles, we check if any two sides are equal:
- \( AB \neq BC \)
- \( AB \neq AC \)
- \( BC \neq AC \)

Since none of the sides are equal, the points do not form an isosceles triangle.

### Final Answer
The points \( (0, 7, 10) \), \( (1, 6, -6) \), and \( (4, 9, -6) \) do not form an isosceles triangle.

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