Question

In $\Delta PQY,\angle P={50}^{\circ }$. If $PQ=3\text{cm}$ and $PY=3.5\text{cm}$, how many such triangles are possible?

• A. Infinite
• B. One
• C. Two
• D. Cannot be determined

Answer 1

The correct option is B One

From the given data, we can say that $\angle P$ is the included angle between sides PQ and PY.
Let us try to construct a triangle with the given measurements.
Steps:
1. Draw a straight line and mark its left endpoint as P.
2. Then set the compass to a width of $3\text{cm}$.
3. Place the pointer head of the compass at P and cut an arc at Q on the line.

4. Now draw a ${50}^{\circ }$ angle with line PQ at point P using a protractor.

5. Set the compass to a width of $3.5\text{cm}$.
6. Place the pointer head of the compass at P and cut an arc at $3.5\text{cm}$ from P. Name this point as Y.

7. Now join points Y and Q.

Thus, we obtained a $\Delta PQY$ of the required measurements and only one unique triangle can be constructed with the given measurements. Even if we change PQ to PY as the base, there will be only one possibility of drawing the unknown third side, i.e., YQ and thus we can get only a unique triangle.
Hence, option C is the correct answer.