Question

Is segment $PQ$ greater than segment $RS$?
I. $PB>RE$, $BQ = ES$.
II. $B$ is a point on $PQ$, $E$ is a point on $RS$.

• A. If the question can be answered with the help of statement I alone.
• B. If the question can be answered with the help of statement II alone.
• C. If both statements I and II are needed to answer the question.
• D. If the question cannot be answered even with the help of both statements.

Hint:

When comparing total lengths from parts, both the part comparisons and their geometric arrangement must be known to combine the inequalities correctly.

Answer 1

The Correct Option is C

From Statement I: We know $PB>RE$ and $BQ = ES$. These are comparisons of parts of $PQ$ and $RS$, but without knowing that $PB$ and $BQ$ are contiguous parts of $PQ$, and $RE$ and $ES$ are contiguous parts of $RS$, we cannot directly add them to compare $PQ$ and $RS$. So Statement I alone is insufficient.

From Statement II: We are told that $B$ lies on $PQ$ and $E$ lies on $RS$, but this only gives us position, not length comparisons. So Statement II alone is also insufficient.

Combining Statements I and II: From Statement II, we confirm $PQ = PB + BQ$ and $RS = RE + ES$. From Statement I, we have $PB>RE$ and $BQ = ES$. Adding these inequalities: \[ PQ = PB + BQ>RE + ES = RS \] Thus $PQ>RS$. Therefore, both statements together are sufficient.