Questions: Find the coordinates of the intersection of the diagonals of square ABCD with vertices A(2,7), B(9,7), C(4,3), and D(-3,3). The coordinates of the intersection of the diagonals are ( , ).

Find the coordinates of the intersection of the diagonals of square ABCD with vertices A(2,7), B(9,7), C(4,3), and D(-3,3).

The coordinates of the intersection of the diagonals are ( , ).

Transcript text: Find the coordinates of the intersection of the diagonals of $\square A B C D$ with vertices $A(2,7)$, $B(9,7), C(4,3)$, and $D(-3,3)$. The coordinates of the intersection of the diagonals are ( $\square$ , $\square$ ).

Solution

Solution Steps

Step 1: Define the Vertices

Let the vertices of the quadrilateral $ \square ABCD $ be defined as follows:

$ A(2, 7) $
$ B(9, 7) $
$ C(4, 3) $
$ D(-3, 3) $

Step 2: Calculate the Midpoint of Diagonal $ AC $

The midpoint $ M_{AC} $ of diagonal $ AC $ is calculated using the formula: \[ M_{AC} = \left( \frac{x_A + x_C}{2}, \frac{y_A + y_C}{2} \right) = \left( \frac{2 + 4}{2}, \frac{7 + 3}{2} \right) = (3.0, 5.0) \]

Step 3: Calculate the Midpoint of Diagonal $ BD $

The midpoint $ M_{BD} $ of diagonal $ BD $ is calculated using the formula: \[ M_{BD} = \left( \frac{x_B + x_D}{2}, \frac{y_B + y_D}{2} \right) = \left( \frac{9 + (-3)}{2}, \frac{7 + 3}{2} \right) = (3.0, 5.0) \]

Step 4: Find the Intersection of the Diagonals

The intersection point $ I $ of the diagonals $ AC $ and $ BD $ is the same as the midpoints of both diagonals: \[ I = \left( \frac{x_{M_{AC}} + x_{M_{BD}}}{2}, \frac{y_{M_{AC}} + y_{M_{BD}}}{2} \right) = (3.0, 5.0) \]