Convex Quadrilateral?

Geometry Level pending

A B C D ABCD is a convex quadrilateral with A B = 3 , B C = 5 AB=3,BC=5 and C D = 7. CD=7. If the bisectors of the four internal angles are concurrent, find the value of A B + B C + C D + D A D A \dfrac{AB+BC+CD+DA}{DA} .


The answer is 4.

Ayush G Rai by Ayush G Rai , 20, India

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1 solution

Ayush G Rai
May 26, 2016

Since the point of concurrence is at the same distance from all four sides, a circle can be drawn having that point as center and touching the sides,as in the figure.Therefore A P = A S , B P = B Q , C R = C Q AP=AS,BP=BQ,CR=CQ and D R = D S . DR=DS. Also the points on the angular bisector are equidistant from the sides.
Adding these four, we get A B + C D = A D + B C ; AB+CD=AD+BC;
i . e . , 3 + 7 = A D + 5. i.e.,3+7=AD+5.
Hence A D = 5. AD=5.
Therefore ( A B + B C + C D + D A ) / D A = 4 . (AB+BC+CD+DA)/DA=\boxed 4.


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