Explanation
Consider the diagram shown in the question.
Given:
AB=AC
∠BOC=100∘
⇒∠BAC=50∘ (angle at the circle is half of angle at the centre by same chord (BC))
Consider the quadrilateral OBAC.
∠BOC (of the quadrilateral) =360∘−100∘=260∘
Consider triangle OBC.
∠OBC=∠OCB (OB and OC are radius)
Similarly, in ΔABC,
∠ABC=∠ACB (Since, AB=AC)
∠ABO+∠OBC=∠ACO+∠OCB
⇒∠ABO=∠ACO
We know that interior angles of a quadrilateral are 360∘. Therefore,
50∘+260∘+∠ABO+∠ACO=360∘
2∠ACO=50∘
∠ACO=25∘
Given- A,B & C are points such that AB=12cm and BC=16cm and BC⊥AB.
To find out- the radius of the circle passing through A,B & C=?
Solution- BC⊥AB i.e ∠ABC=900 and ΔABC is a right one with AC as hypotenuse. ∴AC=√AB2+BC2=√122+162cm=20cm. So the circle passing through A,B & C will have its diameter as AC.
∴ Its radius=12×20cm=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.
∴ Radius, AO=12AD=12×34 cm =17 cm.
∵ON⊥AB
∴ON bisects AB at N and ∠ANO=900
So, AN=12AB=12×30 cm =15 cm
Now, AO=17 cm, AN=15 cm and ∠ANO=900
∴ΔAON is a right one with hypotenuse AO.
So, by pythagoras theorem, we have
AO2−AN2=ON2⇒ON2=(172−152) cm2
⇒ON=8 cm
So, AB is at a distance of 8 cm from the centre O.
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