https://file.aiquickdraw.com/mathbot/file/1729427556829o09mh1s4.jpg

To find the size of \( x \) in this triangle, we can use the **Law of Sines**. This law relates the angles of a triangle to the lengths of its sides. The Law of Sines states:

\[
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
\]

Where:
- \( a, b, c \) are the sides of the triangle.
- \( A, B, C \) are the angles opposite those sides.

### Step-by-step solution:

1. **Label the triangle**:
   - The side opposite the 40° angle is \( x \) (which we are trying to find).
   - The side opposite the 46° angle is 20 cm.
   - The third angle can be found using the fact that the sum of angles in a triangle is always 180°.

2. **Find the third angle**:
   \[
   \text{Third angle} = 180^\circ - 40^\circ - 46^\circ = 94^\circ
   \]

3. **Apply the Law of Sines**:
   Now that we have all three angles, we can use the Law of Sines to find \( x \). Specifically, we will use the ratio between the known side (20 cm) and its opposite angle (46°), and the unknown side \( x \) and its opposite angle (40°):
   \[
   \frac{x}{\sin 40^\circ} = \frac{20 \, \text{cm}}{\sin 46^\circ}
   \]

4. **Solve for \( x \)**:
   Rearranging to isolate \( x \):
   \[
   x = \frac{20 \, \text{cm} \times \sin 40^\circ}{\sin 46^\circ}
   \]

5. **Calculate the values**:
   - \( \sin 40^\circ \approx 0.6428 \)
   - \( \sin 46^\circ \approx 0.7193 \)

   Substituting these values into the equation:
   \[
   x \approx \frac{20 \times 0.6428}{0.7193} \approx \frac{12.856}{0.7193} \approx 17.88 \, \text{cm}
   \]

Thus, the length of \( x \) is approximately **17.88 cm**.

Would you like more details on any step, or do you have any other questions? Here are 5 related questions you might find helpful:

1. How would you solve this if the third angle was given instead?
2. What happens if you know two sides and the angle between them?
3. Can the Law of Sines be applied to any triangle?
4. What other methods exist for solving non-right triangles?
5. How does the Law of Cosines compare to the Law of Sines?

**Tip**: When using the Law of Sines, make sure the angles and sides you use are directly opposite each other!

Find the size of side x in the given triangle with angles 40°, 46°, and a side length of 20 cm opposite the 46° angle.

Learn how to find the size of a missing side in a triangle using the Law of Sines. In this problem, the triangle has angles of 40°, 46°, and one known side of 20 cm. Step-by-step calculations show how to solve for the unknown side. Suitable for Grades 9-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation