Question

If \(A(-3,3)\), \(B(1,1)\), \(C(1,-1)\) and \(D(-2,-2)\) are the vertices of a quadrilateral, the angle between the diagonals \(AC\) and \(BD\) is:

• A. \(\dfrac{\pi}{4}\)
• B. \(\dfrac{\pi}{2}\)
• C. \(\dfrac{\pi}{6}\)
• D. \(\dfrac{\pi}{3}\)

Hint:

For two lines with slopes \(m_1\) and \(m_2\), if \[ m_1m_2=-1, \] then the lines are perpendicular and the angle between them is \(\dfrac{\pi}{2}\).

Answer 1

The Correct Option is B

Step 1: Find the slope of diagonal \(AC\).
The points are \[ A(-3,3) \] and \[ C(1,-1) \] Slope of \(AC\) is \[ m_1=\frac{-1-3}{1-(-3)} \] \[ m_1=\frac{-4}{4} \] \[ m_1=-1 \]

Step 2: Find the slope of diagonal \(BD\).
The points are \[ B(1,1) \] and \[ D(-2,-2) \] Slope of \(BD\) is \[ m_2=\frac{-2-1}{-2-1} \] \[ m_2=\frac{-3}{-3} \] \[ m_2=1 \]

Step 3: Check perpendicularity of diagonals.
Two lines are perpendicular if \[ m_1m_2=-1 \] Here, \[ m_1m_2=(-1)(1) \] \[ m_1m_2=-1 \] Therefore, diagonals \(AC\) and \(BD\) are perpendicular.

Step 4: Find the angle between the diagonals.
Since the diagonals are perpendicular, the angle between them is \[ \frac{\pi}{2} \]

Step 5: Final conclusion.
Hence, the required angle is \[ \boxed{\frac{\pi}{2}} \]