Angle-bisector area split inside a triangle: Triangle PQR has area 180 cm^2. Point S lies on QR, and PS is the angle bisector of ∠QPR. If PQ : PR = 2 : 3, find the area (in cm^2) of triangle PSR.
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A90
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B108
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C144
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D72
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E120
Answer
Correct Answer: 108
Explanation
Introduction / Context:The internal angle bisector divides the opposite side in the ratio of the adjacent sides. Hence, the areas of the two sub-triangles formed on that base are in the same ratio as the base segments.
Given Data / Assumptions:
- Total area(ΔPQR) = 180 cm^2.
- PQ : PR = 2 : 3 and PS is the internal angle bisector of ∠QPR.
Concept / Approach:By the Angle Bisector Theorem, QS : SR = PQ : PR = 2 : 3. Because P is common vertex and the altitude from P to QR is common for ΔPQS and ΔPSR, areas are proportional to their bases on QR.
Step-by-Step Solution:
Let area(ΔPQS) : area(ΔPSR) = QS : SR = 2 : 3.Sum ratio parts = 2 + 3 = 5 → each part = 180/5 = 36.area(ΔPSR) = 3 parts = 3 * 36 = 108 cm^2.Verification / Alternative check:A quick check: the other sub-area is 72; 72 + 108 = 180 matches the total area.
Why Other Options Are Wrong:90, 144, 72, 120 do not equal 3/5 of 180.
Common Pitfalls:Confusing the 2:3 as area(ΔPQS):area(ΔPSR) = 2:3 but then assigning the wrong sub-triangle to the larger part; ensure PSR corresponds to the segment SR (ratio 3).
Final Answer:108