(a) Find the area of the given triangle - Leaving Cert Mathematics - Question 2 - 2016

For the triangle with sides 3 cm, 5 cm, and 7 cm, we will use the Cosine Rule:

$$c^2 = a^2 + b^2 - 2ab \cos(C)$$

Where:

• ( c = 7 , \text{cm} ) (the largest side)
• ( a = 3 , \text{cm} )
• ( b = 5 , \text{cm} )

Substituting the values we get:

$$7^2 = 3^2 + 5^2 - 2(3)(5) \cos(C)$$

This simplifies to:

$$49 = 9 + 25 - 30 \cos(C)$$

Combining and rearranging gives:

$$49 = 34 - 30 \cos(C) \implies \cos(C) = \frac{34 - 49}{30} = \frac{-15}{30} = -\frac{1}{2}$$

Thus, ( C = 120° ), which is indeed the largest angle in the triangle.