An isosceles triangle has a base with a length of 26 inches and base angles that measure $$ 64^{\circ} $$. a. Determine the height of the isosceles triangle to the nearest tenth. $$ \operatorname{Tan}\left(64^{\circ}\right)=\frac{h}{\frac{26}{2}} \rightarrow \frac{h}{13} $$ $$ 13\left(\operatorname{Tan}\left(64^{\circ}\right)\right)=h $$ $$ h=30.5 \mathrm{in} $$ b. Determint the length of the legs of the isosceles triangle to the nearest tenth.

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To solve for the height and the length of the legs of the isosceles triangle, we can follow these steps:

### Part a: Height of the Isosceles Triangle

We already have the equation for the height $$ h $$ based on the tangent of the angle:

$$
h = 13 \cdot \tan(64^{\circ})
$$

Now, we will calculate $$ h $$ using the tangent function.

### Part b: Length of the Legs of the Isosceles Triangle

To find the length of the legs $$ l $$ of the isosceles triangle, we can use the Pythagorean theorem. The legs, the height, and half the base form a right triangle. The relationship can be expressed as:

$$
l = \sqrt{h^2 + \left(\frac{26}{2}\right)^2}
$$

Substituting $$ \frac{26}{2} = 13 $$:

$$
l = \sqrt{h^2 + 13^2}
$$

Now, let's calculate both $$ h $$ and $$ l $$ using the appropriate tool calls.

1. Calculate $$ h $$.
2. Calculate $$ l $$ using the value of $$ h $$.

Let's perform these calculations.
Simplify the expression by following steps:
- step0: Solution:
  $$\sqrt{h^{2}+13^{2}}$$
- step1: Evaluate the power:
  $$\sqrt{h^{2}+169}$$
Calculate or simplify the expression $$ 13*\tan(64) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$13\tan\left(64\right)$$
Calculate or simplify the expression $$ \sqrt(30.522184^2 + 13^2) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\sqrt{30.522184^{2}+13^{2}}$$
- step1: Convert the expressions:
  $$\sqrt{\left(\frac{3815273}{125000}\right)^{2}+13^{2}}$$
- step2: Add the numbers:
  $$\sqrt{\frac{3815273^{2}+169\times 125000^{2}}{125000^{2}}}$$
- step3: Use the properties of radicals:
  $$\frac{\sqrt{3815273^{2}+169\times 125000^{2}}}{\sqrt{125000^{2}}}$$
- step4: Simplify the expression:
  $$\frac{\sqrt{3815273^{2}+169\times 125000^{2}}}{50^{3}}$$
### Part a: Height of the Isosceles Triangle

The height $$ h $$ of the isosceles triangle is calculated as follows:

$$
h = 13 \cdot \tan(64^{\circ} ) \approx 30.5 \text{ inches}
$$

### Part b: Length of the Legs of the Isosceles Triangle

Using the height we found, we can now calculate the length of the legs $$ l $$:

$$
l = \sqrt{h^2 + 13^2} = \sqrt{30.5^2 + 13^2}
$$

Calculating this gives:

$$
l \approx 33.2 \text{ inches}
$$

### Summary of Results
- The height of the isosceles triangle is approximately **30.5 inches**.
- The length of the legs of the isosceles triangle is approximately **33.2 inches**.
The height of the isosceles triangle is approximately 30.5 inches, and the length of each leg is approximately 33.2 inches.

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