a. Prove that the angle bisectors of a convex kite are concurrent as follows. Co

Question

a. Prove that the angle bisectors of a convex kite are concurrent as follows. Consider the kite ABCD in the figure below and its diagonal BD. Prove that line BD is on the angle bisector of angle B as well as angle D. Then construct the angle bisector of angle A and point O where that angle bisector intersects line BD. Next let dsub1 be the distance from O to line AD, dsub2 be the distance from O to line AB, dsub3 be the distance from O to line BC, and dsub4 be the distance from O to line DC, Argue that dsub1=dsub2, dsub2=dsub3, and dsub4=dsub1. Conclude that dsub3=dsub4 and hence that O is on the angle bisector of angle C.

b. Construct any kite and the circle inscribed in the kite. Clearly identify the radius of the circle. Describe the construction and prove that it is valid.

Explanation / Answer

a kite is a quadrilateral with two pairs of adjacent sides equal

therefore, in triangle BCD and triangle BAD

BA = BC ( adjacent sides of a kite)

AD = CD ( adjacent sides of a kite )

BD common

so triangle BCD is congruent to triangle BAD

hence , angle CDB = angle ADB

and angle ABD = angle CBD

so we can conclude that BD is the angle bisector of angle B and D

to inscribe a circle within a kite

draw the axis of symmetry which is BD in our case

bisect one of the non vertex angles (C or A ) and extend this line so that it meets line BD at any point say o'

now drop a perpendicular from o' to any of the 4 sides of the kite at b'

then taking o' as centre and o'b' as radius inscribe a circle within the kite

all kites are tangential quadrilaterals that is they are 4 sided figures into which a circle can be inscribed such that each side will touch the circle at any one point

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