Explanation
Consider the diagram shown in the question.
Given:
AB=AC
\angle BOC=100{}^\circ
\Rightarrow \angle BAC=50{}^\circ (angle at the circle is half of angle at the centre by same chord \left( BC \right))
Consider the quadrilateral OBAC.
\angle BOC (of the quadrilateral) =360{}^\circ -100{}^\circ =260{}^\circ
Consider triangle OBC.
\angle OBC=\angle OCB (OB and OC are radius)
Similarly, in \Delta ABC,
\angle ABC=\angle ACB (Since, AB=AC)
\angle ABO+\angle OBC=\angle ACO+\angle OCB
\Rightarrow \angle ABO=\angle ACO
We know that interior angles of a quadrilateral are 360{}^\circ . Therefore,
50{}^\circ +260{}^\circ +\angle ABO+\angle ACO=360{}^\circ
2\angle ACO=50{}^\circ
\angle ACO=25{}^\circ
Given- A, B & C are points such that AB=12cm and BC=16cm and BC\bot AB.
To find out- the radius of the circle passing through A, B & C=?
Solution- BC\bot AB i.e \angle ABC={ 90 }^{ 0 } and \Delta ABC is a right one with AC as hypotenuse. \therefore AC=\sqrt { { AB }^{ 2 }+{ BC }^{ 2 } } =\sqrt { { 12 }^{ 2 }+{ 16 }^{ 2 } } cm=20cm. So the circle passing through A, B & C will have its diameter as AC.
\therefore Its radius=\dfrac { 1 }{ 2 } \times 20cm=10cm.
Ans- Option C.
It is given that, Diameter, AD = 34 cm and chord, AB = 30 cm
Draw a perpendicular ON from O to AB. It meets AB at N.
\therefore Radius, AO=\dfrac { 1 }{ 2 } AD=\dfrac { 1 }{ 2 } \times 34 cm =17 cm.
\because ON\bot AB
\therefore ON bisects AB at N and \angle ANO=90^0
So, AN=\dfrac { 1 }{ 2 }AB=\dfrac { 1 }{ 2 } \times 30 cm =15 cm
Now, AO=17 cm, AN=15 cm and \angle ANO=90^0
\therefore \Delta AON is a right one with hypotenuse AO.
So, by pythagoras theorem, we have
{ AO }^{ 2 }-{ AN }^{ 2 }={ ON }^{ 2 }\Rightarrow { ON }^{ 2 }={ (17 }^{ 2 }{ -15 }^{ 2 })\ { cm }^{ 2 }
\Rightarrow ON=8 cm
So, AB is at a distance of 8 cm from the centre O.
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