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 .

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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:

$$
\boxed{51 \text{ cm}}
$$
The diagonals $$ AC $$ and $$ BD $$ are both 51 cm long.

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 .

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