Question

ABCD is a square. The bisector of ∠DBC cuts AC and CD at E and F respectively. Prove that BF × CE = BE × DF.

Answer 1

Answer:

BF×CE=BE×DF has it is a square while we bisect it will be equal

Answer 2

Answer:

A/q

BF is the bisector of angle DBC [given]

i.e. angle DBF= angle CBF

We know that the angles of a square is 90° which is bisected by its diagonals hence becomes 45° each.

Now as BF is intersecting angle DBC the angles are now 22.5° each also angle BDC = angle BCE = 45°.

•°• BDE~ BCE  [ AA similarity]

by one theorem we also know that ratio of the areas of two similar triangle is equal to the ratio of the squares of their corresponding sides.

so, (DF/CE)^2=(BF/BE)^2

=> DF/CE= BF/BE= DF*BE= BF*CE proved.

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