To solve for $x$, we use the formula for the area of a triangle:

$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}.$

Here:

• The area is $35 \, \text{cm}^2$,
• The base is $x + 8$,
• The height is $5 \, \text{cm}$.

Substituting these values into the area formula:

$35 = \frac{1}{2} \times (x + 8) \times 5.$

Step 1: Simplify the equation

$35 = \frac{5}{2} \times (x + 8).$

Multiply both sides by 2 to eliminate the fraction: $70 = 5(x + 8).$

Step 2: Expand and solve for $x$

$70 = 5x + 40.$

Subtract 40 from both sides: $30 = 5x.$

Divide both sides by 5: $x = 6.$

Final Answer:

$x = 6.$

Would you like a step-by-step explanation of how the triangle formula works or have further questions?

Questions to Explore Further:

1. How is the area formula of a triangle derived?
2. Can this approach be used for non-right triangles?
3. What happens if $x$ results in a negative length? Is it valid?
4. How can we verify the solution for $x$?
5. Can we use a similar method for solving areas of trapezoids?

Tip:

Always check your solution by substituting $x$ back into the equation to ensure it satisfies the problem!