Prove,if the diagonals of a c cyclic quadrilateral are perpendicular to each other show that the line passing through the point of intersection of diagonals and midpoint of a side is perpendicular to the opposite side
Answers
Proof :
Given:
(i) ◻ ABCD is a cyclic quadrilateral
(ii) Diagonals AC and BD intersect at M and AC⊥BD
(iii) M is the mid point of AB
To Prove:
MN ⊥ DC
Proof:
In right angled triangle AOB, AM is the median
∴ OM = AM = MB = 1/2 AB …. (property of median on the hypotenuse)
∴ ∠MBO = ∠MOB (1)
& ∠MAO = ∠MOA …. (Isosceles triangle theorem) …. (2)
∠BOA = 90° ….. (given)
∴ ∠MOB + ∠MOA = 90° ….(3)
∠MOA = ∠CON ….. (vertically opposite angles) …..(4)
∠ABD = ∠ACD ….. (angles subtended by the same chord)
i.e. ∠MBO = ∠OCN …..(5)
∴ ∠MOB = ∠ OCN ….. from (1) and (5) .... (6)
∴ ∠OCN + ∠CON = 90° ….. from (3), (4) and (6) ..... (7)
∴ ∠OND = ∠OCN + ∠CON ….. (Exterior angle of triangle ONC)
= 90 ….. from (7)
∴ MN ⊥ DC
Answer:
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Step-by-step explanation:
