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: Identify the diagonals.
The quadrilateral has vertices $A(-3,3)$, $B(1,1)$, $C(1,-1)$, $D(-2,-2)$. The diagonals are $AC$ and $BD$.

Step 2: Find the direction vector of $AC$.
\[ \vec{AC} = C - A = (1-(-3),\ -1-3) = (4, -4). \]

Step 3: Find the direction vector of $BD$.
\[ \vec{BD} = D - B = (-2-1,\ -2-1) = (-3, -3). \]

Step 4: Compute the dot product.
\[ \vec{AC} \cdot \vec{BD} = (4)(-3) + (-4)(-3) = -12 + 12 = 0. \]

Step 5: Conclude the angle.
Since the dot product is zero, the two diagonals are perpendicular. The angle between them is $\dfrac{\pi}{2}$.

Step 6: State the answer.
The angle between the diagonals $AC$ and $BD$ is $\dfrac{\pi}{2}$.
\[ \boxed{\dfrac{\pi}{2}} \]