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

Step 1: Understand what congruence requires.
Two triangles are congruent only if all their corresponding sides and angles match exactly, not just because some overall measure like area happens to match.
Step 2: Test Statement I alone.
Statement I says the areas of triangle ABC and triangle PQR are equal. Two triangles can easily have equal area without being congruent; for example, a triangle with base 6 and height 4 has the same area (12) as a triangle with base 8 and height 3, yet these triangles have completely different side lengths and are not congruent. So Statement I alone is not sufficient.
Step 3: Test Statement II alone.
Statement II says both triangles are right-angled. Being right-angled says nothing about the actual side lengths; a right triangle with legs 3 and 4 is nothing like a right triangle with legs 5 and 12, so this alone cannot establish congruence either. Statement II alone is not sufficient.
Step 4: Test both statements together.
Combine equal area with both being right triangles. A right triangle with legs 3 and 4 has area 1/2 x 3 x 4 = 6. A right triangle with legs 2 and 6 has area 1/2 x 2 x 6 = 6 as well. Both are right triangles with the same area, yet their side lengths (3, 4, 5) versus (2, 6, sqrt40) are completely different, so they are not congruent. This single counterexample shows that even both pieces of information together fail to guarantee congruence.
Step 5: Conclusion.
Since neither statement alone, nor both together, can guarantee that triangle ABC is congruent to triangle PQR, the data in both statements together is not sufficient to answer the question. This matches option (4).