In triangle PQR, q = 8 cm, r = 6 cm and cos P = $\frac{1}{12}$. Calculate the value of p.

Analysis

To find the side length $p$ in triangle $PQR$ when two sides ($q$ and $r$) and the included angle ($P$) are known, we use the Cosine Rule.

Concept: The Cosine Rule

The Cosine Rule states that for any triangle $PQR$ with sides $p, q,$ and $r$ opposite to angles $P, Q,$ and $R$ respectively:

This formula is essential for solving triangles when you have a "Side-Angle-Side" (SAS) configuration.

Step-by-Step Solution

1. Identify the given values:

• $q = 8\text{ cm}$
• $r = 6\text{ cm}$
• $\cos P = \frac{1}{12}$

2. Substitute the values into the Cosine Rule formula:

3. Perform the arithmetic:

• Calculate the squares: $8^2 = 64$ and $6^2 = 36$
• Calculate the product: $2 \times 8 \times 6 = 96$
• Multiply by $\cos P$: $96 \times \frac{1}{12} = 8$

4. Combine the results:

5. Solve for $p$:

Conclusion

Comparing our result to the given options, we find that $\sqrt{92}$ cm corresponds to option C.

Exam Tip

Always remember that the Cosine Rule is essentially a generalized version of Pythagoras' Theorem. If the angle $P$ were $90^\circ$, $\cos 90^\circ$ would be $0$, and the formula would simplify perfectly to $p^2 = q^2 + r^2$. Keeping this connection in mind helps you remember the formula during the heat of the exam!