Step 1: Using the distance formula to find the lengths of the sides.
To check whether the points form a right-angled triangle, we need to find the distances between each pair of points. The distance formula between two points \( (x_1, y_1, z_1) \) and \( (x_2, y_2, z_2) \) in three-dimensional space is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \] Let the points be:
- \( A = (4, 4, 2) \)
- \( B = (3, 5, 2) \)
- \( C = (-1, -1, 2) \)
We will now calculate the distances between these points.
(i) Distance between \( A \) and \( B \) (\( AB \)):
\[ AB = \sqrt{(3 - 4)^2 + (5 - 4)^2 + (2 - 2)^2} = \sqrt{(-1)^2 + (1)^2 + 0^2} = \sqrt{1 + 1} = \sqrt{2} \] (ii) Distance between \( B \) and \( C \) (\( BC \)):
\[ BC = \sqrt{(-1 - 3)^2 + (-1 - 5)^2 + (2 - 2)^2} = \sqrt{(-4)^2 + (-6)^2 + 0^2} = \sqrt{16 + 36} = \sqrt{52} = 2\sqrt{13} \] (iii) Distance between \( C \) and \( A \) (\( CA \)):
\[ CA = \sqrt{(-1 - 4)^2 + (-1 - 4)^2 + (2 - 2)^2} = \sqrt{(-5)^2 + (-5)^2 + 0^2} = \sqrt{25 + 25} = \sqrt{50} = 5\sqrt{2} \] Step 2: Verifying the Pythagorean theorem.
For these points to form a right-angled triangle, the Pythagorean theorem must hold. That is, for the three sides \( a \), \( b \), and \( c \), where \( c \) is the hypotenuse, the relationship \( a^2 + b^2 = c^2 \) must be satisfied.
We have the following side lengths:
- \( AB = \sqrt{2} \)
- \( BC = 2\sqrt{13} \)
- \( CA = 5\sqrt{2} \)
Now, we check if the Pythagorean theorem holds by comparing the sum of the squares of the two smaller sides to the square of the longest side:
\[ AB^2 + BC^2 = (\sqrt{2})^2 + (2\sqrt{13})^2 = 2 + 52 = 54 \] \[ CA^2 = (5\sqrt{2})^2 = 50 \] Clearly, \( AB^2 + BC^2 \neq CA^2 \), so the longest side isn't the hypotenuse. Now let's check the other two combinations:
\[ AB^2 + CA^2 = (\sqrt{2})^2 + (5\sqrt{2})^2 = 2 + 50 = 52 \] \[ BC^2 = (2\sqrt{13})^2 = 52 \] Since \( AB^2 + CA^2 = BC^2 \), the Pythagorean theorem holds, and therefore the points \( A \), \( B \), and \( C \) form a right-angled triangle with the right angle at \( A \).
Final Answer: The points \( (4, 4, 2) \), \( (3, 5, 2) \), and \( (-1, -1, 2) \) are the vertices of a right-angled triangle.