https://file.aiquickdraw.com/mathbot/file/1727010477138epjujnt2.png

To solve for \( x \) in this triangle, we can approach the problem by analyzing the right triangles formed by the height of the triangle, which splits it into two smaller right triangles.

### Step-by-step approach:

1. **Label the height as \( h \)**:
   The blue dashed line represents the height of the triangle, and it splits the base into two parts. Since the two base angles are \( 35^\circ \) and \( 42^\circ \), the height divides the base into two segments.

   - The segment adjacent to the \( 35^\circ \) angle is \( a \).
   - The segment adjacent to the \( 42^\circ \) angle is \( b \).

2. **Use trigonometry**:
   We can use trigonometric ratios (sine, cosine, and tangent) in both right triangles to express \( h \), \( a \), and \( b \).

   From the triangle with the \( 35^\circ \) angle:
   \[
   \tan(35^\circ) = \frac{h}{a} \implies a = \frac{h}{\tan(35^\circ)}
   \]

   From the triangle with the \( 42^\circ \) angle:
   \[
   \tan(42^\circ) = \frac{h}{b} \implies b = \frac{h}{\tan(42^\circ)}
   \]

3. **Base relation**:
   The sum of the segments \( a \) and \( b \) equals the total base length, which is 39 units.
   \[
   a + b = 39
   \]
   Substituting the expressions for \( a \) and \( b \):
   \[
   \frac{h}{\tan(35^\circ)} + \frac{h}{\tan(42^\circ)} = 39
   \]

4. **Solve for \( h \)**:
   Factor out \( h \) from the equation:
   \[
   h \left( \frac{1}{\tan(35^\circ)} + \frac{1}{\tan(42^\circ)} \right) = 39
   \]
   Now solve for \( h \):
   \[
   h = \frac{39}{\left( \frac{1}{\tan(35^\circ)} + \frac{1}{\tan(42^\circ)} \right)}
   \]

5. **Find \( x \)**:
   Finally, use one of the right triangles to find \( x \). For example, from the triangle with the \( 35^\circ \) angle:
   \[
   \sin(35^\circ) = \frac{h}{x} \implies x = \frac{h}{\sin(35^\circ)}
   \]

Let's calculate these values.The length of side \( x \) is approximately \( 26.78 \) units, and the height of the triangle is approximately \( 15.36 \) units.

Would you like further details or have any additional questions?

Here are some related questions to expand your understanding:
1. How would the solution change if the base of the triangle were different?
2. How can you find the area of this triangle given the height?
3. What is the significance of using trigonometric ratios in solving triangles?
4. How would you solve for the height if only the side \( x \) were given?
5. Can you generalize this approach for any isosceles triangle?

**Tip**: Always verify trigonometric ratios by ensuring that the angle measures and calculations are accurate for precise results.

Solve for x in the isosceles triangle with angles 35°, 42°, and a base of 39.

Learn how to solve for side x in an isosceles triangle with base 39, angles 35° and 42°. The solution involves using trigonometric ratios to find the height and the length of the side x. Suitable for high school students (Grades 10-12).

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation