In a rectangle ABCD , the length of the diagonal \( \overline{A C} \) is given by \( (18 x+15) \mathrm{cm} \) and the length of diagonal \( \overline{B D} \) is given by \( (20 \mathrm{x}+11) \mathrm{cm} \). Find actual lengths of AC and BD .

In a rectangle, the diagonals are equal in length. Therefore, we can set the expressions for the lengths of the diagonals equal to each other: \[ 18x + 15 = 20x + 11 \] Now, let's solve this equation step by step. 1. **Rearranging the equation**: We will move all terms involving \( x \) to one side and constant terms to the other side. \[ 18x - 20x = 11 - 15 \] 2. **Simplifying**: \[ -2x = -4 \] 3. **Dividing by -2**: \[ x = 2 \] Now that we have the value of \( x \), we can substitute it back into either expression for the diagonals to find their actual lengths. Let's calculate the length of diagonal \( AC \): \[ AC = 18x + 15 = 18(2) + 15 \] Calculating this gives: \[ AC = 36 + 15 = 51 \text{ cm} \] Now, let's calculate the length of diagonal \( BD \): \[ BD = 20x + 11 = 20(2) + 11 \] Calculating this gives: \[ BD = 40 + 11 = 51 \text{ cm} \] Thus, the actual lengths of the diagonals \( AC \) and \( BD \) are both: \[ \boxed{51 \text{ cm}} \]