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.)

To find the length of \( AD \) in the triangle, we can apply the **Law of Cosines** or use the fact that this is a particular triangle with symmetry, given that both sides \( u = v = 16 \) and both angles \( \angle x = \angle y = 25^\circ \).

### Steps:

1. **Recognize symmetry**: Since \( u = v = 16 \) and \( \angle x = \angle y = 25^\circ \), the triangle appears to be **isosceles**. This symmetry allows us to simplify the approach.

2. **Use Law of Cosines**: The Law of Cosines for a triangle states:
   \[
   c^2 = a^2 + b^2 - 2ab \cos(C)
   \]
   where \( c \) is the side opposite angle \( C \), and \( a \) and \( b \) are the other two sides.

3. **Apply it to this triangle**: Let \( AD \) be the side opposite to the angle between the equal sides (angle \( 25^\circ \)). Thus, we can calculate \( AD \) as follows:

\[
AD^2 = u^2 + v^2 - 2uv \cos(\angle A)
\]

Since \( u = v = 16 \) and \( \angle A = 25^\circ \), substitute these values:

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

\[
AD^2 = 256 + 256 - 2(16)(16) \cos(25^\circ)
\]

Now let's compute \( AD \). I'll calculate it for you.The length of \( AD \) is approximately \( 6.93 \) units.

Would you like more details or have any questions?

Here are 5 related questions for further practice:
1. How would the calculation change if \( u \) and \( v \) were not equal?
2. What is the length of the third side if \( \angle z = 130^\circ \) in the triangle?
3. How do you derive the Law of Cosines from basic trigonometric principles?
4. What would the area of this triangle be given its dimensions?
5. Can you find the altitude from \( A \) to \( BD \) using the same triangle?

**Tip**: When two angles and two sides of a triangle are equal, the Law of Cosines simplifies calculations for the unknown side lengths, especially in isosceles triangles.

Solve a triangle problem to find the length of AD using the Law of Cosines. Given u = 16, v = 16, and ∠x = ∠y = 25°, calculate AD to two decimal places. Learn key trigonometry concepts, including isosceles triangle properties and the Law of Cosines.

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