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We are given a parallelogram \(ABCD\) with the following side lengths:

- \(AB = (8x - 5)\) inches
- \(BC = (3x + 10)\) inches

The perimeter of the parallelogram is \(46\) inches. The perimeter of a parallelogram is calculated as:

\[
P = 2(AB + BC)
\]

Given \(P = 46\), we can set up the following equation:

\[
46 = 2[(8x - 5) + (3x + 10)]
\]

Let's solve this step-by-step.

### Step 1: Expand and simplify the equation
\[
46 = 2[(8x - 5) + (3x + 10)]
\]
First, simplify the terms inside the parentheses:
\[
(8x - 5) + (3x + 10) = 8x + 3x - 5 + 10 = 11x + 5
\]
Now substitute back:
\[
46 = 2(11x + 5)
\]

### Step 2: Divide both sides by 2
\[
23 = 11x + 5
\]

### Step 3: Solve for \(x\)
Subtract 5 from both sides:
\[
23 - 5 = 11x
\]
\[
18 = 11x
\]
Now, divide by 11:
\[
x = \frac{18}{11} \approx 1.64
\]

### Step 4: Calculate the length of \(DA\) (which is the same as \(AB\) in a parallelogram)
Substitute \(x = \frac{18}{11}\) into the expression for \(AB\):
\[
DA = AB = 8x - 5 = 8\left(\frac{18}{11}\right) - 5
\]
\[
DA = \frac{144}{11} - 5 = \frac{144}{11} - \frac{55}{11} = \frac{89}{11} \approx 8.09
\]

Thus, \(DA \approx 8\) inches.

The correct answer is \( \boxed{8} \) inches.

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Would you like more details, or do you have any questions?

Here are 5 related questions:
1. What are the properties of a parallelogram that help us solve problems like this?
2. How would the solution change if the perimeter were different?
3. Can you derive the general formula for the perimeter of a quadrilateral?
4. What happens to the perimeter if the lengths of opposite sides are not equal?
5. How can we check if the given parallelogram is a square based on side lengths?

**Tip:** For any parallelogram, opposite sides are equal in length, which helps simplify calculations when finding perimeters or solving for unknowns.

The perimeter of parallelogram ABCD is 46 inches. What is DA? Given AB = (8x - 5) inches and BC = (3x + 10) inches.

Solve a parallelogram perimeter problem where AB = (8x - 5) inches, BC = (3x + 10) inches, and the perimeter is 46 inches. Learn step-by-step how to find DA using algebraic methods.

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