To find the length of side $p$ in $\Delta PQR$, we will apply the Law of Sines, which states:
$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

Step 1: Identify the given values

• $r$: the circumradius of the triangle, $r = 38 \, \text{cm}$.
• $\angle P = 49^\circ$, $\angle Q = 127^\circ$.
• $\angle R = 180^\circ - \angle P - \angle Q = 180^\circ - 49^\circ - 127^\circ = 4^\circ$.

In a triangle with circumradius $r$, any side $a$, $b$, or $c$ can be found using: $a = 2r \cdot \sin A$ $b = 2r \cdot \sin B$ $c = 2r \cdot \sin C$

Here, $p$ is the side opposite $\angle P$, so: $p = 2r \cdot \sin P$

Step 2: Substitute the known values

$p = 2(38) \cdot \sin(49^\circ)$

Step 3: Calculate

1. Compute $\sin(49^\circ)$: $\sin(49^\circ) \approx 0.7547$
2. Substitute into the formula: $p = 2(38)(0.7547)$ $p \approx 57.3 \, \text{cm}$

Final Answer:

$p \approx 57 \, \text{cm} \, (\text{to the nearest centimeter}).$

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Would you like further details or explanations about any step?

5 Related Questions:

1. How does the Law of Sines work for non-right triangles?
2. What is the circumradius of a triangle, and how is it used?
3. How do we calculate the angles in a triangle if only two angles are given?
4. Can the circumradius formula be derived from the Law of Sines?
5. What happens to side lengths when the circumradius increases?

Tip:

Always ensure the sum of the angles in a triangle is $180^\circ$ to avoid calculation errors.