Consider the non-right triangle shown below that has interior angle measures of 1.25 rad, 0.368 rad, and 1.523 rad. The two non-horizontal legs have lengths of 1.55 and 4.085 cm.

A non-right triangle is drawn with angle measures of 1.25 radians, 0.368 radians, and 1.523 radians. The side opposite the 1.25 radian angle measures 4.085 centimeters, the side opposite the 0.368 radian angle measures 1.55 centimeters, and the side opposite the 1.523 radian angle has an unknown length of x centimeters.
What is the length of the horizontal leg of the triangle, in centimeters?

x
=

5.5
0.23
4.3
1.47

To solve for the length of the horizontal leg of the triangle, we can use the **Law of Sines**, which states:

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

Given:
- Angle \( A = 1.25 \) radians, opposite side \( a = 4.085 \) cm
- Angle \( B = 0.368 \) radians, opposite side \( b = 1.55 \) cm
- Angle \( C = 1.523 \) radians, opposite side \( c = x \) cm

We can find \( x \) using:

\[
\frac{x}{\sin(1.523)} = \frac{4.085}{\sin(1.25)}
\]

Solving for \( x \):

\[
x = \frac{4.085 \times \sin(1.523)}{\sin(1.25)}
\]

Let's compute the value of \( x \):

- \(\sin(1.523) \approx 0.998\)
- \(\sin(1.25) \approx 0.948\)

Now:

\[
x = \frac{4.085 \times 0.998}{0.948} \approx \frac{4.07783}{0.948} \approx 4.3 \text{ cm}
\]

So, the length of the horizontal leg \( x \) is **4.3 cm**.

Would you like any more details or have further questions?

Here are 5 related questions:
1. How can you use the Law of Cosines to solve for the sides in a non-right triangle?
2. How does the Law of Sines apply when dealing with ambiguous cases?
3. Can you find the area of this triangle using Heron's formula?
4. How do you convert angle measurements from radians to degrees and vice versa?
5. What is the significance of the triangle inequality theorem in this scenario?

**Tip:** Always double-check the units (degrees vs. radians) when working with trigonometric functions to avoid errors in calculations.

Explore how to find the length of the horizontal leg in a non-right triangle with given angle measures and side lengths using the Law of Sines. Detailed step-by-step solution included. Suitable for Grades 10-12.

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