For the triangle shown, find the length *AD*. (Assume *u* = 16, *v* = 16, ∠*x* = 25°, and ∠*y* = 25°. Round your answer to two decimal places.)
*AD* =

To find the length of \( AD \), we can use the Law of Cosines in the triangle. Let's assume the given information refers to a triangle with the following details:

- \( AB = u = 16 \)
- \( AC = v = 16 \)
- \( \angle A = x = 25^\circ \)
- \( \angle B = y = 25^\circ \)

### Key Approach:
Since \( AB = AC \), this is an **isosceles triangle**. The angle at \( A \) is \( 25^\circ \), and the remaining angles at \( B \) and \( C \) will both be \( 25^\circ \). Therefore, the triangle is isosceles and symmetrical.

### Solution:
We need to apply the Law of Cosines to calculate the length of \( AD \). The Law of Cosines is:

\[
c^2 = a^2 + b^2 - 2ab \cdot \cos(C)
\]

Here:

- \( a = b = u = v = 16 \)
- \( C = \angle x = 25^\circ \)

\[
AD^2 = 16^2 + 16^2 - 2 \cdot 16 \cdot 16 \cdot \cos(25^\circ)
\]

Let's calculate this.The length of \( AD \) is approximately \( 6.93 \) units, rounded to two decimal places.

Would you like further details or clarification?

Here are some related questions you might want to explore:
1. What is the Law of Cosines and when is it applicable?
2. How would the solution change if the angle \( x \) were different?
3. Can you solve the problem using the Law of Sines?
4. How do you find angles in an isosceles triangle?
5. What happens to the triangle if \( u \neq v \)?

**Tip:** Always ensure your angles are in the correct units (radians or degrees) when using trigonometric functions in formulas.

For the triangle shown, find the length AD. (Assume u = 16, v = 16, ∠x = 25°, and ∠y = 25°. Round your answer to two decimal places.) AD =

Discover how to calculate the length of AD in an isosceles triangle with given side lengths and angles using the Law of Cosines. This problem involves trigonometric principles and provides a step-by-step solution suitable for students in Grades 10-12.

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