TANCET 2025 Solved Papers

Concept:To decide whether a triangle is right angled or acute angled, the given information must determine the angles or the relation among the sides.Explanation:Statement (1) gives only the area, $140\ \mathrm{cm}^2$, and two sides $a$ and $b$. The area is expressed as $\frac12 ab\sin C = 140$. Since actual values of $a$ and $b$ are not known, this relation can hold for many different triangles, including acute, right, and obtuse triangles. So statement (1) alone is not sufficient.Statement (2) says median $AD$ to side $BC$ equals altitude $AE$ to side $BC$. Since $AE$ is the shortest distance from $A$ to the line $BC$, equality forces $D$ and $E$ to be the same point. Hence $AD$ is the perpendicular bisector of $BC$, giving $AB=AC$. Thus the triangle is isosceles. But an isosceles triangle can be right angled or acute angled, depending on the apex angle. Statement (2) alone is not sufficient.Combining both statements, the area condition and the isosceles condition still do not fix the actual side lengths or the angles. Therefore, additional data are needed.Answer:Option D — Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.