CBSE - X - Mathematics
if PT is drawn parallel to chord AB in a circle with centre O .prove that APB is an isoceles triangle
Construction: Join PO and produce it to D.
Now, OP is perpendicular TP (tangent makes a 90° angle with the radius of the circle at the point of contact)
Also, TP is parallel to AB (given)
angle ADP = 90° (interior angles)
So, OD is perpendicular to AB.
Since, a perpendicular drawn from the center of the circle to a chord bisects it,
PD is a bisector of AB. i.e. AD = DB
Now in triangle ADP and BDP
AB = DB (proved above)
angle ADP = angle BDP (both are 90°)
PD = DP (common)
Triangle ADP Triangle BDP (By SAS congruence criterion)
Hence, angle PAD = angle PBD (By CPCT)
Thus, APB is an isosceles triangle.
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