Question

Triangle ABC and Triangle PQR are congruent
(I) Area of Triangle ABC and Triangle PQR are same
(II) Triangle ABC and Triangle PQR are right angle Triangles

• A. Answer (1) if data in Statement I alone is sufficient to answer the question but the data in Statement II alone is not sufficient to answer the question.
• B. Answer (2) if data in Statement II alone is sufficient to answer the question but the data in Statement I alone is not sufficient to answer the question.
• C. Answer (3) if data in Statement I and II together are necessary to answer the question.
• D. Answer (4) if data in Statement I and II together are not sufficient to answer the question.

Hint:

Equal area or being right triangles does not fix side lengths; try to find a counterexample.

Answer 1

The Correct Option is D

Congruence between two triangles means every side and every angle of one triangle matches the corresponding side and angle of the other. Equal area and equal triangle type (right-angled) are much weaker conditions than this, so the real question is whether area plus right-angled-ness pins down the full shape. It does not, and a single counterexample settles this cleanly.

Consider two right triangles with legs $(p,q)$ chosen so that $\frac12 pq$ is the same number both times:

• Triangle A: legs $3$ and $4$, area $= \frac12(3)(4) = 6$, hypotenuse $= 5$.
• Triangle B: legs $2$ and $6$, area $= \frac12(2)(6) = 6$, hypotenuse $= \sqrt{40} = 2\sqrt{10}$.

Both triangles are right-angled (satisfying Statement II) and both have area $6$ (satisfying Statement I), yet their side lengths $3,4,5$ and $2,6,2\sqrt{10}$ do not match at all. So even with both statements assumed true simultaneously, triangle ABC and triangle PQR could be shaped like Triangle A and Triangle B above: same area, both right triangles, but not congruent.

Since a valid counterexample survives even when both statements are combined, neither statement alone, nor the two together, is enough to establish congruence. The correct choice is that the data in Statements I and II together are \(\boxed{\text{not sufficient}}\), which is option (4).