Question

In a regular pentagon PQRST, prove that △PRS is an isosceles triangle​

Answer 1

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Each angle of a regular pentagon is 108 degrees. In triangle PTS, angle T = 108 degrees, and PT = TS. So, angle TPS = angle TSP = 36 degrees.

Similarly, angle QPR = 36 degrees. So, angle SPR = 108 – (36 + 36) = 36 degrees

Also, PS = PR

Area of triangle PSR = ½ * PS * PR * sin 36

Area of the pentagon = Sum of areas of the three triangles = Area of triangle PTS + Area of triangle PSR + Area of triangle PQR

= ½ * PT * PS * sin (angle TPS) + ½ * PS * PR * sin 36 + ½ * PQ * PR * sin (angle QPR)

= ½ * PT * PS * sin 36 + ½ * PS * PR * sin 36 + ½ * PQ * PR * sin 36

Required ratio = ½ * PS * PR * sin 36 / (½ * PT * PS * sin 36 + ½ * PS * PR * sin 36 + ½ * PQ * PR * sin 36)

= PR / (PT + PR + PQ)

PT = PQ = SR

SR = 2 * PR cos (angle PSR) = 2 * PR * cos 72

Ratio = PR / (2 * PR cos 72 + PR + 2 * PR cos 72) = 1 / (4 cos 72 + 1)

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Answer 2

Answer:

4cos72+1 is the correct answer

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