In ΔPQR, r = 38 cm, mm∠P=49° and mm∠Q=127°. Find the length of p, to the nearest centimeter.


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

---

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.

In ΔPQR, r = 38 cm, mm∠P=49° and mm∠Q=127°. Find the length of p, to the nearest centimeter.

Find the length of side p in triangle ΔPQR where r = 38 cm, ∠P = 49°, and ∠Q = 127°. Learn how to apply the Law of Sines and circumradius formula with step-by-step calculations. Suitable for Grades 10-12.

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