Rectangle Diagonals C79474

1. **Problem Statement:** We are given a rectangle $ABCD$ with vertices $A$ (top-left), $B$ (top-right), $C$ (bottom-right), and $D$ (bottom-left). Both diagonals $AC$ and $BD$ are drawn and intersect at the center. 2. **Properties of a Rectangle and its Diagonals:** - Opposite sides are equal and parallel: $AB = DC$ and $AD = BC$. - All angles are right angles ($90^\circ$). - The diagonals are equal in length: $AC = BD$. - The diagonals bisect each other, meaning they intersect at their midpoints. 3. **Formula for Diagonal Length:** If the length of the rectangle is $l$ and the width is $w$, then the length of each diagonal $d$ is given by the Pythagorean theorem: $$d = \sqrt{l^2 + w^2}$$ 4. **Intersection Point of Diagonals:** Since the diagonals bisect each other, the intersection point $O$ is the midpoint of both $AC$ and $BD$. If $A = (x_1, y_1)$ and $C = (x_3, y_3)$, then $$O = \left(\frac{x_1 + x_3}{2}, \frac{y_1 + y_3}{2}\right)$$ Similarly for $B = (x_2, y_2)$ and $D = (x_4, y_4)$, $$O = \left(\frac{x_2 + x_4}{2}, \frac{y_2 + y_4}{2}\right)$$ 5. **Summary:** - The diagonals are equal in length. - They bisect each other at the center point $O$. - The rectangle's properties ensure right angles and parallel opposite sides. This completes the analysis of the rectangle and its diagonals as described in the graph.