Question

${\textbf{Subject : Mathematics}}$

${\textbf{Chapter : Tangency}}$


A point P is 16 cm from the centre of the circle. The length of tangent drawn from P to the circle is 12 cm. Find the radius of the circle.​

Answer 1

Hey !

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ɢɪᴠᴇɴ : ᴘ ɪs ᴀ ᴘᴏɪɴᴛ ᴀᴛ ᴅɪsᴛᴀɴᴄᴇ ᴏғ 16ᴄᴍ ғʀᴏᴍ ᴛʜᴇ ᴄᴇɴᴛʀᴇ ᴏғ ᴀ ᴄɪʀᴄʟᴇ ᴏ. ᴛʜᴇ ʟᴇɴɢᴛʜ ᴏғ ᴛᴀɴɢᴇɴᴛ ᴅʀᴀᴡɴ ғʀᴏᴍ ᴘ ᴛᴏ ᴛʜᴇ ᴄɪʀᴄʟᴇ ɪs 12 ᴄᴍ.

ɪᴛ ɪs ʀᴇǫᴜɪʀᴇᴅ ᴛᴏ ᴍᴇᴀsᴜʀᴇ ᴛʜᴇ ʀᴀᴅɪᴜs ᴏғ ᴛʜᴇ ᴄɪʀᴄʟᴇ.

ɴᴏᴡ, ∠ᴏᴛᴘ= 90°

( ∵ ᴀ ᴛᴀɴɢᴇɴᴛ ᴛᴏ ᴀ ᴄɪʀᴄʟᴇ ɪs ⊥ ᴛᴏ ᴛʜᴇ ʀᴀᴅɪᴜs ᴛʜʀᴏᴜɢʜ ᴛʜᴇ ᴘᴏɪɴᴛ ᴏғ ᴄᴏɴᴛᴀᴄᴛ).

∴ ɪɴ ʀɪɢʜᴛ ᴀɴɢʟᴇ ᴛʀɪᴀɴɢʟᴇ ᴏᴛᴘ,

ᴏᴘ² = ᴏᴛ²+ ᴛᴘ²

ᴏʀ, 16² = ᴏᴛ² + (12)²

ᴏʀ, 256-144 = ᴏᴛ²

ᴏʀ, 112 = ᴏᴛ²

∴ ᴏᴛ = 10.89 ᴄᴍ

ʜᴇɴᴄᴇ, ʀᴀᴅɪᴜs = 10.89 ᴄᴍ

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Thanks !

Answer 2

Given:

P is a point at distance of 16cm from the centre of a circle. The length of tangent drawn from P to the circle is 12cm.

It is required to measure the radius of a circle.

Now, angle OTP = 90°

(Since a tangent to a circle is perpendicular to the radius through the point of contact).

so in right triangle OTP,

OP^2= OT^2+TP^2

or, 16^2 = OT^2+ 12^2

256= OT^2+144

256-144= OT^2

OT^2= 112

so, OT = 10.89cm.

I hope it will help you.