In this problem, we have a right triangle with the following information:

• Angle $63^\circ$ at the base left corner.
• Angle $39^\circ$ at the base right corner.
• The length of the height $a = 56$, perpendicular to the base $x$.
• We are tasked with finding the base $x$, rounded to the nearest hundredth.

Approach:

We can use trigonometric functions to solve this. Since $a$ is the opposite side to the angle $63^\circ$ and $x$ is the adjacent side, we can use the tangent function:

$\tan(63^\circ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{a}{x}$

Substituting the known values:

$\tan(63^\circ) = \frac{56}{x}$

Solving for $x$:

$x = \frac{56}{\tan(63^\circ)}$

Let me compute this.The length of the base $x$ is approximately $28.53$, rounded to the nearest hundredth.

Do you want more details or have any questions about the solution?

Here are 5 related questions to expand your understanding:

1. How would you solve for $x$ if $a$ were unknown and only the angles and $x$ were given?
2. What trigonometric function would you use if the angle given was at the base of the triangle?
3. How can you check if your solution for $x$ is correct using another trigonometric function?
4. What is the relation between the angles and sides in a right triangle?
5. How could the Pythagorean theorem be applied if both legs were known?

Tip: When using trigonometric functions, always ensure your calculator or software is set to the correct mode (degrees or radians) based on the problem’s given angle units.