In triangle PQR, the area is 180 cm². Point S lies on side QR such that PS is the internal angle bisector of angle QPR. If PQ : PR = 2 : 3, what is the area, in square centimetres, of triangle PSR?

Introduction / Context:
This question uses the angle bisector theorem in a triangle and applies it to areas. When a segment from a vertex bisects the angle at that vertex, it divides the opposite side in the ratio of the adjacent sides. Since the triangles on each side of the bisector share a common altitude from the vertex, their areas are also in the same ratio. This allows us to partition the total area based on a given side length ratio.

Given Data / Assumptions:

• Triangle PQR has total area 180 cm².
• S lies on side QR.
• PS is the internal angle bisector of angle QPR.
• The side ratio is PQ : PR = 2 : 3.
• We must find the area of triangle PSR.
• All geometry is in the Euclidean plane and the standard angle bisector theorem applies.

Concept / Approach:
The angle bisector theorem states: If PS is the internal angle bisector of ∠QPR, then QS / SR = PQ / PR. Since PQ : PR = 2 : 3, this implies QS : SR = 2 : 3. The triangles P-Q-S and P-S-R share the same altitude from P to side QR, so their areas are proportional to their bases QS and SR. Therefore, area(ΔPQS) : area(ΔPSR) = QS : SR = 2 : 3. The total area of triangle PQR is the sum of these two areas.

Step-by-Step Solution:
Step 1: Use the angle bisector theorem. Given PQ : PR = 2 : 3 and PS is the angle bisector at P. Therefore, QS : SR = 2 : 3. Step 2: Relate side ratio to area ratio. Triangles P-Q-S and P-S-R share vertex P and have bases QS and SR along QR. Since the altitude from P to QR is common, area is proportional to the base: area(ΔPQS) : area(ΔPSR) = QS : SR = 2 : 3. Step 3: Use the total area of triangle PQR. Let area(ΔPQS) = 2k and area(ΔPSR) = 3k. Then total area 2k + 3k = 5k. Given 5k = 180 cm², so k = 180 / 5 = 36. Step 4: Compute the desired area. area(ΔPSR) = 3k = 3 * 36 = 108 cm².

Verification / Alternative check:
We can also compute area(ΔPQS) as 2k = 72 cm². Adding both sub areas gives 72 + 108 = 180 cm², which matches the given area of triangle PQR. Thus the partition is consistent with the total area and the ratio 2 : 3, confirming that 108 cm² is correct for triangle PSR.

Why Other Options Are Wrong:
Option 1: 72 sq cm corresponds to the smaller region area(ΔPQS), not the area of ΔPSR which is larger. Option 2: 90 sq cm would correspond to a 1 : 1 ratio of areas (90 + 90 = 180), which contradicts the given side ratio 2 : 3. Option 4: 144 sq cm implies the other triangle has area 36 sq cm, giving an area ratio 36 : 144 = 1 : 4, not equal to the side ratio 2 : 3. Option 5: None of these is incorrect because 108 sq cm is present as an option and matches the correct calculation.

Common Pitfalls:
A typical mistake is to assume that the angle bisector divides the triangle into two equal areas, which is only true if the adjacent sides are equal. Another error is to invert the ratio and assign 3k to the smaller region and 2k to the larger region without checking that PQ is the smaller side. Some students may misapply the angle bisector theorem and use QS : SR = PR : PQ instead of PQ : PR. Carefully linking side ratios to segment ratios and then to area ratios prevents these errors.

Final Answer:
The area of triangle PSR is 108 sq cm.