Properties Of Triangles

The cosine rule is the standard JEE Advanced tool for relating all three sides of a triangle to a single angle, generalizing the Pythagorean theorem to triangles that need not contain a right angle. The angle opposite the side $b=14$ is angle $B$, and the rule states $\cos B=\frac{a^2+c^2-b^2}{2ac}$, where the side opposite the target angle appears with a minus sign in the numerator. Substituting the squared sides $a^2=169$, $c^2=225$, and $b^2=196$, the numerator becomes $169+225-196=198$, while the denominator is $2\cdot 13\cdot 15=390$. Dividing both numerator and denominator by their common factor six gives $\cos B=\frac{198}{390}=\frac{33}{65}$. Option $\frac{5}{13}$ corresponds to a different angle in the triangle and a mismatched ratio. Option $\frac{4}{5}$ comes from misapplying the rule to the side $a$ rather than $b$. Option $\frac{12}{13}$ mistakes a sine-type value for a cosine. This matches the canonical cosine-rule derivation used repeatedly in JEE Advanced triangle questions. As a final plausibility check, $\frac{33}{65}\approx 0.508$ is a legitimate positive cosine, indicating an acute angle, which is fully consistent with the well-known fact that the 13-14-15 triangle is acute.