Maharashtra Class 10 Answered
Prove- If the diagonals of a cyclic quadrilateral are perpendicular to each other show that the line passing through the point of intersection of diagonals and mid point of a side is perpendicular to the opposite side.
Asked by Ampra1952 | 07 Dec, 2019, 10:22: AM
Expert Answer
If the diagonals of a cyclic quadrilateral are perpendicular to each other,
show that the line passing through the point of interaction of the diagonals and the midpoint of a side, is perpendicular to the opposite side
Let T be the given midpoint of PQ,
Now in a circle line joining the midpoint of chord and center is perpendicular to the chord
so angle PTO is 90 degree
if we take triangles PTO and QYO, we can prove them congruent by SAS
so angle TPO and angle TQO are congruent by C.P.C.T.
similarly angle POT and QOT
so
angle POT + angle QOT = 90.....given that diagonals are perpendicular
angle POT = angle QOT = 45
in triangle PTO
angle POT = 45
angle PTO = 90
so
angle TPO = 45
hence
angle TPO = angle TQO = 45
now
angle OSU= angle TPO = 45... angles subtended by same arc
angle TQO = angle ORU = 45... angles subtended by same arc
angle TOQ = angle SOU = 45... vertically opp. angles
angle POT = angle ROU = 45... vertically opp. angles
now in triangle OSU
angle OSU= 45
angle SOU = 45
hence
angle OUS = 90...sum of all angles in triangle
hence proved
Answered by Arun | 07 Dec, 2019, 03:23: PM
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