Question

Prove that (4, 4, 2), (3, 5, 2) and (-1, -1, 2) are vertices of a right angle triangle.

Hint:

For problems involving proving geometric properties like right angles, using the dot product of vectors is often faster and less calculation-intensive than using the distance formula and the Pythagorean theorem. A dot product of zero immediately confirms perpendicularity.

Answer 1

Step 1: Understanding the Concept:
A triangle is a right-angled triangle if two of its sides are perpendicular to each other. In vector terms, this means the dot product of the vectors representing those two sides is zero. Alternatively, we can use the distance formula to find the lengths of the three sides and check if they satisfy the Pythagorean theorem \( (a^2 + b^2 = c^2) \).
Step 2: Key Formula or Approach:
Let the given vertices be A(4, 4, 2), B(3, 5, 2), and C(-1, -1, 2).
1. Find the vectors representing the sides of the triangle: \( \vec{AB} \), \( \vec{BC} \), and \( \vec{AC} \).
- \( \vec{AB} = B - A = (x_2-x_1, y_2-y_1, z_2-z_1) \)
2. Calculate the dot product of pairs of these vectors. If any dot product is zero, the corresponding vectors are perpendicular.
- \( \vec{u} \cdot \vec{v} = u_1v_1 + u_2v_2 + u_3v_3 \)
3. If a dot product is zero, the triangle is right-angled.
Step 3: Detailed Explanation or Calculation:
Step 3.1: Define the vertices.
Let A = (4, 4, 2), B = (3, 5, 2), C = (-1, -1, 2).
Step 3.2: Find the side vectors.
\[ \vec{AB} = B - A = (3-4, 5-4, 2-2) = (-1, 1, 0) \] \[ \vec{BC} = C - B = (-1-3, -1-5, 2-2) = (-4, -6, 0) \] \[ \vec{AC} = C - A = (-1-4, -1-4, 2-2) = (-5, -5, 0) \] Step 3.3: Calculate the dot products.
Let's check if any pair of vectors is perpendicular.
Dot product of \( \vec{AB} \) and \( \vec{AC} \):
\[ \vec{AB} \cdot \vec{AC} = (-1)(-5) + (1)(-5) + (0)(0) \] \[ = 5 - 5 + 0 = 0 \] Since the dot product of \( \vec{AB} \) and \( \vec{AC} \) is zero, the vectors are perpendicular. This means the angle between side AB and side AC is 90 degrees.
Alternative Method (Using Pythagorean Theorem):
Calculate the square of the lengths of the sides.
\[ |\vec{AB}|^2 = (-1)^2 + 1^2 + 0^2 = 1 + 1 = 2 \] \[ |\vec{BC}|^2 = (-4)^2 + (-6)^2 + 0^2 = 16 + 36 = 52 \] \[ |\vec{AC}|^2 = (-5)^2 + (-5)^2 + 0^2 = 25 + 25 = 50 \] Check if the sum of the squares of the two smaller sides equals the square of the largest side:
\[ |\vec{AB}|^2 + |\vec{AC}|^2 = 2 + 50 = 52 \] This is equal to \( |\vec{BC}|^2 \).
Since \( |\vec{AB}|^2 + |\vec{AC}|^2 = |\vec{BC}|^2 \), the Pythagorean theorem holds.
Step 4: Final Answer:
Both methods confirm that the triangle formed by the vertices A, B, and C is a right-angled triangle, with the right angle at vertex A.