APQR in which QR = 42.2 cm, PQ == 21.2 cm, and LP == 100.5°

To solve the triangle \(PQR\) with the given dataTo solve the triangle \(PQR\) with the given 
- \(QR = 42.2 \, \text{cm}\) (side opposite to \(\angle P\))
- \(PQ = 21.2 \, \text{cm}\) (side opposite to \(\angle R\))
- \(\angle P = 100.5^\circ\)

We need to determine the remaining angle \(\angle Q\) and \(\angle R\), as well as the side \(PR\).

### Step 1: Use the Law of Sines
The Law of Sines states:

\[
\frac{QR}{\sin P} = \frac{PR}{\sin Q} = \frac{PQ}{\sin R}
\]

We know \(\angle P\), \(QR\), and \(PQ\). So, we can find the unknown angle \(\angle R\) first.

\[
\frac{QR}{\sin P} = \frac{PQ}{\sin R}
\]

### Step 2: Solve for \(\sin R\)
\[
\sin R = \frac{PQ \times \sin P}{QR}
\]
\[
\sin R = \frac{21.2 \times \sin(100.5^\circ)}{42.2}
\]

Calculate:

\[
\sin(100.5^\circ) \approx 0.9848
\]
\[
\sin R = \frac{21.2 \times 0.9848}{42.2} \approx \frac{20.86816}{42.2} \approx 0.4944
\]

Thus,

\[
R \approx \arcsin(0.4944) \approx 29.6^\circ
\]

### Step 3: Determine \(\angle Q\)
Using the angle sum property of a triangle:

\[
\angle Q = 180^\circ - \angle P - \angle R
\]
\[
\angle Q = 180^\circ - 100.5^\circ - 29.6^\circ = 49.9^\circ
\]

### Step 4: Calculate the side \(PR\) using the Law of Sines
\[
\frac{PR}{\sin Q} = \frac{QR}{\sin P}
\]
\[
PR = \frac{QR \times \sin Q}{\sin P}
\]
\[
PR = \frac{42.2 \times \sin(49.9^\circ)}{\sin(100.5^\circ)}
\]

Calculate:

\[
\sin(49.9^\circ) \approx 0.766
\]
\[
PR = \frac{42.2 \times 0.766}{0.9848} \approx \frac{32.3352}{0.9848} \approx 32.8 \, \text{cm}
\]

### Summary:
- \(\angle R \approx 29.6^\circ\)
- \(\angle Q \approx 49.9^\circ\)
- \(PR \approx 32.8 \, \text{cm}\)

Would you like more details or have any questions?

### Related Questions:
1. How can you verify these results using the Law of Cosines?
2. What would the area of triangle \(PQR\) be using Heron's formula?
3. How can you find the altitude from \(P\) to side \(QR\)?
4. What is the inradius of triangle \(PQR\)?
5. How does the obtuse angle \(\angle P\) affect the triangle's properties?

### Tip:
In triangles with an obtuse angle, always ensure to use the correct sine values and carefully consider the relationships between the angles and sides.

Learn how to solve triangle PQR with given sides and angles: QR = 42.2 cm, PQ = 21.2 cm, and angle LP = 100.5 degrees. Apply the Law of Sines and angle sum property of a triangle to find angles R and Q, and side PR. Suitable for high school students.

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