Mathematics

Problem

When the area

\Delta

of a triangle with known sides

a

and

b

and unknown third side

c

is specified, application of the area formula leads to a quadratic equation in

c

. a. (i) In not more than four lines, explain why the equation obtained is quadratic in

c

. b. (ii) Give two distinct geometrical reasons why, after solving the quadratic equation, one of the roots may have to be rejected while choosing the value of

c

Intuitive explanation

The key insight is that the area formula $\Delta = \frac{1}{2}ab\sin C$ connects the unknown side $c$ to the area through the angle $C$ opposite it. Using the law of cosines to express $\cos C$ in terms of $c$ , then deriving $\sin C$ from it, you can substitute into the area formula and rearrange—this creates a quadratic equation in $c$ because $\sin C$ depends on $c^2$ . This is a powerful technique for finding a missing side when you know two sides and the area instead of an angle.

Intuitive explanation

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