Give all the theorem of circle of class 10
Answers
Theorem 1
The tangent at any point of a circle is perpendicular to the radius through the point of contact.
Construction: Draw a circle with centre O. Draw a tangent XY which touches point P at the circle.
10 circle of theorem 1
To Prove: OP is perpendicular to XY.
Draw a point Q on XY; other than O and join OQ. Here OQ is longer than the radius OP.
OQ > OP
For every point on the line XY other than O, like Q1, Q2, Q3, ……….Qn;
OQ1>OP
OQ2>OP
OQ3>OP
OQ4>OP
Since OP is the shortest line
Hence, OP ⊥ XY proved
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Circle
Theorems on Circle
Tangent to a circle: A line which intersects a circle at any one point is called the tangent.
There is only one tangent at a point of the circle.
The tangent to a circle is perpendicular to the radius through the point of contact.
The lengths of the two tangents from an external point to a circle are equal.
Theorem 2
The lengths of tangents drawn from an external point to a circle are equal.
Construction: Draw a circle with centre O. From a point P outside the circle, draw two tangents P and R.
10 circle theorem 1
To Prove: PQ = PR
Proof: In Δ POQ and Δ POR
O
Q
=
O
R
(radii)
P
O
=
P
O
(common side)
∠
P
Q
O
=
∠
P
R
O
(Right angle)
Hence;
Δ
P
O
Q
≅
Δ
P
O
R
proved
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- First circle theorem - angles at the centre and at the circumference.
- Second circle theorem - angle in a semicircle.
- Third circle theorem - angles in the same segment.
- Fourth circle theorem - angles in a cyclic quadlateral.
- Fifth circle theorem - length of tangents.
- Sixth circle theorem - angle between circle tangent and radius.
- Seventh circle theorem - alternate segment theorem.
- Eighth circle theorem - perpendicular from the centre bisects the chord