To find the length of each of the two equal sides of the isosceles triangle, we can follow these steps:

Step 1: Divide the Isosceles Triangle

Divide the isosceles triangle into two right triangles by drawing a perpendicular line from the vertex opposite the base to the midpoint of the base. This perpendicular line bisects the base and creates two right triangles.

Step 2: Identify the Right Triangle Dimensions

In each right triangle:

• The base of the right triangle is half of the original base of the isosceles triangle.
• The angle adjacent to the base is one of the base angles of the isosceles triangle.

Given:

• The length of the base $b = 49 \frac{23}{100}$ inches (converted to a decimal, this is approximately $49.23$ inches).
• Each base angle $\theta = 31.35^\circ$.

Step 3: Calculate the Half-Base

The half-base of the right triangle is: $\text{Half-Base} = \frac{49.23}{2} = 24.615 \text{ inches}$

Step 4: Use the Tangent Function

In the right triangle, the tangent of the base angle is the ratio of the opposite side (half-base) to the adjacent side (which is the height of the original triangle).

Let $h$ be the height of the triangle.

$\tan(31.35^\circ) = \frac{24.615}{h}$

$h = \frac{24.615}{\tan(31.35^\circ)}$

Using a calculator to compute $h$:

$h = \frac{24.615}{0.6093} \approx 40.39 \text{ inches}$

Step 5: Calculate the Length of Each Equal Side

Now that we know the height $h$, we can calculate the length of each equal side $s$ using the Pythagorean theorem:

$s = \sqrt{h^2 + \left(\frac{b}{2}\right)^2}$

Substitute the values:

$s = \sqrt{(40.39)^2 + (24.615)^2}$

$s = \sqrt{1631.3521 + 605.971225}$

$s = \sqrt{2237.323325} \approx 47.31 \text{ inches}$

Conclusion:

The length of each of the two equal sides of the isosceles triangle is approximately 47.31 inches.

Would you like more details or have any questions?

Here are some related questions you might find interesting:

1. How would the length of the equal sides change if the base angle increased?
2. What would happen to the triangle's height if the base angle decreased to 25 degrees?
3. How can we calculate the area of this isosceles triangle?
4. What would be the perimeter of this isosceles triangle?
5. Can you determine the coordinates of the vertices of this triangle if the base is aligned along the x-axis?

Tip: Always double-check your angle mode (degrees/radians) when using trigonometric functions on a calculator.