prove that the perpendicular at the point of contact to a tangent to a circle
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Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre.
Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre.
Byju's Answer Standard X Mathematics
Tangent Perpendicular to Radius at Point of Contact
Prove that th... Question
Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre.
Open in App Solution
STEP :Proof
Let us assume a circle with centre
O and let AB
be the tangent intersecting the circle at point
P .
Also let us assume a point
X such that XP is perpendicular to AB .
STEP 2 : Proving thatXP
passes through centreO We know that
Tangent of a circle is perpendicular to radius at point of contact
⇒OP⊥AB
(Tangent at any point of circle is perpendicular to the radius through point of contact)
So, ∠OPB=90° ...(1)
we have already assumed that
XP is perpendicular to AB ∠XPB=90° ...(2) Now from equation (1) and (2) ∠OPB=∠XPB=90°
This condition is possible only if line
XP passes through O . Since, XP
passes through centre
O .
Therefore, it is proved that the perpendicular at the point of contact to the tangent of a circle passes through the centre.
Suggest Corrections 42
SIMILAR QUESTIONS
Q.Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre.
Q.State true or false:
For a tangent, the perpendicular line from the point of contact to the circle passes through the centre.
Q. Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contactQ. Prove that tangent drawn at any point of a circle is perpendicular to the radius through the point of contact.Q. Question 5Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre.
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Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre of the circle.
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Question
Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre of the circle.
Medium Open in App Solution Verified by Toppr
Given a circle with center O and AB the tangent intersecting circle at point P
and prove that OP⊥AB
We know that tangent of the circle is perpendicular to radius at points of contact Hence
OP⊥AB So, ∠OPB=90 o ..........(i)
Now lets assume some point X
Such that XP⊥AN Hence ∠XPB=90 o .........(ii) From eq (i) & (ii) ∠OPB=∠XPB=90 o
Which is possible only if line XP passes though O
Hence perpendicular to tangent passes though centre
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Ex 10.2, 5
Ex 10.2,5 Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre. Given: Let us assume a circle with centre O & AB be the tangent intersecting circle at point P To prove: OP AB Proof: We know that Tangent of circle is perpendicular t
Ex 10.2, 5 - Chapter 10 Class 10 Circles (Term 2)
Last updated at Aug. 5, 2021 by Teachoo
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Transcript
Ex 10.2,5 Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre. Given: Let us assume a circle with centre O & AB be the tangent intersecting circle at point P To prove: OP AB Proof: We know that Tangent of circle is perpendicular to radius at point of contact Hence, OP AB So, OPB = 90 Now lets assume some point X , such that XP AB Hence, XPB = 90 From (1) and (2) OPB = XPB = 90 Which is possible only if line XP passes through O Hence , perpendicular to tangent passes through centre
Next: Ex 10.2, 6 →Ask a doubt (live) Facebook Whatsapp
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Davneet Singh
Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.
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