Question

Verify that (0,7,10), (-1,6,6) and (-4,9,6) are the vertices of a right angled triangle.

Hint:

To verify if three points form a right angled triangle, check if the Pythagorean theorem holds by comparing the squared distances between the points.

Answer 1

Step 1: Calculate the squared distances between the points.
We are given the points \( P_1(0, 7, 10) \), \( P_2(-1, 6, 6) \), and \( P_3(-4, 9, 6) \). First, we calculate the squared distances between the points: \[ d_{12}^2 = (0 - (-1))^2 + (7 - 6)^2 + (10 - 6)^2 = 1^2 + 1^2 + 4^2 = 1 + 1 + 16 = 18 \] \[ d_{13}^2 = (0 - (-4))^2 + (7 - 9)^2 + (10 - 6)^2 = 4^2 + (-2)^2 + 4^2 = 16 + 4 + 16 = 36 \] \[ d_{23}^2 = (-1 - (-4))^2 + (6 - 9)^2 + (6 - 6)^2 = 3^2 + (-3)^2 + 0^2 = 9 + 9 + 0 = 18 \]
Step 2: Check if the points satisfy the Pythagorean theorem.
Now, check if the Pythagorean theorem holds: \[ d_{12}^2 + d_{23}^2 = 18 + 18 = 36 = d_{13}^2 \] Since the sum of the squares of two sides equals the square of the third side, the points form a right angled triangle.
Step 3: Conclusion.
The points \( (0, 7, 10) \), \( (-1, 6, 6) \), and \( (-4, 9, 6) \) are the vertices of a right angled triangle.