In triangle PQR, the side lengths are p = 8\text{ cm} and q = 15\text{ cm}, and the includ… — JAMB UTME

In triangle $PQR$ , the side lengths are $p = 8\text{ cm}$ and $q = 15\text{ cm}$ , and the included angle $\angle R = 60^\circ$ . What is the length of side $r$ in centimeters?

Answer: 13 cm

Answer

The length of side

r

13\text{ cm}

Using the Cosine Rule formula

r^2 = p^2 + q^2 - 2pq \cos R

with

p=8

q=15

, and

R=60^\circ

, we calculate

r^2 = 64 + 225 - 240(0.5) = 169

. Taking the square root gives

r = 13\text{ cm}

Step-by-Step Solution

Identify the given values and appropriate trigonometric rule

Givens:

p = 8\text{ cm}

q = 15\text{ cm}

\angle R = 60^\circ

. Since two sides and the included angle (SAS) are given, use the Cosine Rule:

r^2 = p^2 + q^2 - 2pq \cos R

The Cosine Rule is required to find the third side when two sides and their included angle are known.

Substitute the values into the Cosine Rule formula

r^2 = 8^2 + 15^2 - 2(8)(15) \cos 60^\circ

Replacing variables with their numerical equivalents sets up the algebraic calculation.

Calculate the terms and evaluate

r^2

r^2 = 64 + 225 - 240 \times 0.5 = 289 - 120 = 169

Since

\cos 60^\circ = 0.5

, simplify the arithmetic operations.

Solve for side length

r

r = \sqrt{169} = 13\text{ cm}

Take the square root of both sides to obtain the length of side

r

Key Concept

Applying the Cosine Rule to find an unknown side given two sides and the included angle (SAS)

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