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In the given figure, if OA = 5 cm, AB = 8 cm and OD is perpendicular to AB then what is the length of CD?

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2 cm

3 cm

4 cm

5 cm

answer is A.

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Detailed Solution

Given, in the given figure, if OA = 5 cm, AB = 8 cm and OD is perpendicular to AB.
Question Image

OC is perpendicular on chord AB.

∴ OC

bisects the chord AB.

\Rightarrow AC = CB

Now,

AC + CB = AB

\Rightarrow AC + CB = 8

\Rightarrow AC = \frac{8}{2}

\Rightarrow AC = 4

△ OCA

is a right angled triangle.
Pythagoras theorem states that the square of the hypotenuse side in a right-angled triangle equals the sum of the squares of the other two sides.
According to the pythagorean theorem,

∴ A O^{2} = A C^{2} + O C^{2}

\Rightarrow 5^{2} = 4^{2} + O C^{2}

\Rightarrow 5^{2} - 4^{2} = O C^{2}

\Rightarrow O C^{2} = 9

\Rightarrow OC = 3

cm
Since, OD is the radius of the circle.

∴ OA = OD = 5

CD = OD - OC

\Rightarrow 5 - 3 = 2

cm
Hence, the length of the CD is 2 cm.
Hence, option 1 is correct.

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