Questions: Given: Rectangle ABCD Prove: AC bisects BD, and BD bisects AC. (a/2, b/2)

Transcript text: Given: Rectangle $A B C D$ Prove: $\overline{A C}$ bisects $\overline{B D}$, and $\overline{B D}$ bisects $\overline{A C}$. $\left(\frac{a}{2}, \frac{b}{2}\right)$

Solution

Solution Steps

Step 1: Find the midpoint of AC

The coordinates of A are (0, 0) and the coordinates of C are (2a, 2b). The midpoint formula is ((x₁ + x₂)/2, (y₁ + y₂)/2). Midpoint of AC = ((0 + 2a)/2, (0 + 2b)/2) = (a, b)

Step 2: Find the midpoint of BD

The coordinates of B are (0, 2b) and the coordinates of D are (2a, 0). Midpoint of BD = ((0 + 2a)/2, (2b + 0)/2) = (a, b)

Step 3: Conclusion

Since the midpoints of both diagonals are the same, (a, b), the diagonals bisect each other.