Question

In the given figure, O is the center of the circle, ${\text{OA}} = 10{\text{ cm}}$ and ${\text{OC}} = 6{\text{ cm}}$, the length of AB is:

A.$10{\text{ cm}}$
B.${\text{8 cm}}$
C.$12{\text{ cm}}$
D.$16{\text{ cm}}$

Answer 1

Hint: We will apply the Pythagoras theorem in the triangle OCA to get the value of AC and hence multiply it by 2 to get the value of AB because we know that the perpendicular from the center of circle bisects the chord.

Complete step by step answer:
A figure is given in which O is the center of the circle and the values of OA is given as $10{\text{ cm}}$ and the value of OC is given as $6{\text{ cm}}$ and we have to find the length of chord AB.
As it can be seen in the given figure that OC is perpendicular drawn to O on line AB so $\angle {\text{C = 90}}^\circ $ .
We will apply the Pythagoras theorem in the triangle OCA to get the value of AC because we have the values of the other two sides of the triangle and also $\angle {\text{C = 90}}^\circ $.
According to the Pythagoras theorem,
${{\text{H}}^2} = {{\text{B}}^2} + {{\text{P}}^2}$
Where, H represents the hypotenuse of triangle B represents the base of the triangle and P represents the perpendicular of the triangle.
In $\Delta {\text{OCA}}$ by Pythagoras theorem,
${\text{A}}{{\text{O}}^{\text{2}}}{\text{ = A}}{{\text{C}}^{\text{2}}}{\text{ + O}}{{\text{C}}^{\text{2}}}$
Now, substitute the values of AO and OC in the Pythagoras theorem to find the AC.
${10^2} = {\text{A}}{{\text{C}}^2} + {6^2}$
$100 = {\text{A}}{{\text{C}}^2} + 36$
Now, we solve the equation for the value of $AC$.
$100 - 36 = {\text{A}}{{\text{C}}^{\text{2}}}$
$64 = {\text{A}}{{\text{C}}^{\text{2}}}$
$\sqrt {64} = {\text{AC}}$
${\text{8}} = {\text{AC}}$
Hence, the AC is equal to $8\,{\text{cm}}$ .
Now, we have the value of AC and as we know the perpendicular from the center of the circle to bisect the chord into two equal parts which means,
${\text{AB}} = 2 X AC$
Because ${\text{AC}} = {\text{BC}}$
And we have the value of AC, now we will substitute the value of AC.
${\text{AB}} = 2 X AC$
${\text{AB}} = 2 \times 8$
${\text{AB}} = 16{\text{ cm}}$
Hence, the length of AB is $16{\text{ cm}}$ and option D is correct.


Note:
Remember that the perpendicular drawn from the center of the circle to the chord always bisects the chord in two equal parts, thus we can conclude that $AC = BC$ in the given problem.