ABC in which BC = 62.5 cm, LA = 112°, and 4C = 42°

To solve the triangle \(ABC\) with the given dataTo solve the triangle \(ABC\) with the given 
- \(BC = 62.5 \, \text{cm}\)
- \(\angle A = 112^\circ\)
- \(\angle C = 42^\circ\)

### Step 1: Determine \(\angle B\)
Using the angle sum property of a triangle, the sum of the angles in any triangle is always \(180^\circ\).

\[
\angle B = 180^\circ - \angle A - \angle C
\]

\[
\angle B = 180^\circ - 112^\circ - 42^\circ = 26^\circ
\]

### Step 2: Use the Law of Sines
The Law of Sines states:

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

Where:
- \(a\) is the side opposite to angle \(A\),
- \(b\) is the side opposite to angle \(B\),
- \(c\) is the side opposite to angle \(C\).

Here, \(c = BC = 62.5 \, \text{cm}\).

We can find the other sides using the Law of Sines.

#### Finding side \(AB = c\) opposite to \(\angle C\):
\[
\frac{AB}{\sin C} = \frac{BC}{\sin A}
\]
\[
AB = \frac{BC \times \sin C}{\sin A}
\]
\[
AB = \frac{62.5 \times \sin(42^\circ)}{\sin(112^\circ)}
\]

Let's calculate:

\[
AB \approx \frac{62.5 \times 0.6691}{0.9272} \approx \frac{41.819}{0.9272} \approx 45.1 \, \text{cm}
\]

#### Finding side \(AC = b\) opposite to \(\angle B\):
\[
\frac{AC}{\sin B} = \frac{BC}{\sin A}
\]
\[
AC = \frac{BC \times \sin B}{\sin A}
\]
\[
AC = \frac{62.5 \times \sin(26^\circ)}{\sin(112^\circ)}
\]

Let's calculate:

\[
AC \approx \frac{62.5 \times 0.4384}{0.9272} \approx \frac{27.4}{0.9272} \approx 29.5 \, \text{cm}
\]

### Summary:
- \(\angle B = 26^\circ\)
- \(AB \approx 45.1 \, \text{cm}\)
- \(AC \approx 29.5 \, \text{cm}\)

Would you like more details or have any questions?

### Related Questions:
1. How would you find the area of triangle \(ABC\)?
2. What is the circumradius of triangle \(ABC\) given these sides?
3. How can you verify the sides calculated using another method?
4. How does the Law of Cosines compare to the Law of Sines in solving this triangle?
5. What are the properties of the triangle given these angles and sides?

### Tip:
Always double-check your calculations, especially when dealing with trigonometric functions, to avoid rounding errors.

Learn how to solve triangle ABC with given side BC = 62.5 cm, angle LA = 112°, and angle 4C = 42° using the Law of Sines and angle sum property of a triangle. Detailed step-by-step solution provided.

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