Page 9 - Lesson note 5 Angle subtended by arc Ch.- 10 Circle

In ΔOBD and ΔOCD:
OB = OC [Radius of same circle]
0
ODB = ODC [Each 90
OD = OD [Common]
Therefore, By RHS Condition
ΔOBD ≅ ΔOCD
So, BOD = COD….. (i)[By CPCT]
Again,
BOC = 2 BAC The angle subtended by an arc at the centre is double the angle
Subtended by it at any point on the remaining part of the circle.)
2 BOD = 2 BAC [Using(i)]
BOD = BAC
Hence proved.

Question 6:Two circles intersect at two points A and B. AD and AC are diameters to the two
circles. Prove that B lies on the line segment DC.

Solution : Join AB.
ABD = 90° (Angle in a semicircle)
ABC = 90° (Angle in a semicircle)
So, ABD + ABC = 90° + 90° = 180°
Therefore, DBC is a line. That is B lies on the line segment DC.