Important Formulas: Circles

Circle

• A circle is formed by the aggregation of points in a plane that share a consistent distance from a specific fixed point.

Centre

• The center of the circle is referred to as the fixed point.

• The distance that remains consistent from the center is termed the radius.

Chord

• A line segment joining any two points on a circle is called a chord.

Diameter

• A chord passing through the centre of the circle is called diameter. It is the longest chord.

Tangent

When a line meets the circle at one point or two coincidings The line is known as points, a tangent.
The tangent to a circle is perpendicular to the radius through the point of contact.
⇒ OP ⊥ AB

The lengths of the two tangents from an external point to a circle are equal.
⇒ AP = PB
Length of Tangent Segment
PB and PA are normally called the lengths of tangents from outside point P.

Properties of Tangent to Circle

Theorem 1: Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact.
Given: XY is a tangent at point P to the circle with centre O.
To prove: OP ⊥ XY
Construction: Take a point Q on XY other than P and join OQ
Proof: If point Q lies inside the circle, then XY will become a secant and not a tangent to the circle
OQ > OP

This happens with every point on line XY except point P. OP is the shortest of all the distances of point O the points of XY
OP ⊥ XY …[Shortest side is the perpendicular]

Theorem 2: A line drawn through the endpoint of a radius and perpendicular to it, is tangent to the circle.
Given: A circle C(O, r) and a line APB is perpendicular to OP, where OP is the radius.
To prove: AB is tangent at P.
Construction: Take a point Q on line AB, different from P, and join OQ.
Proof: Since OP ⊥ AB
OP < OQ ⇒ OQ > OP

Point Q lies outside the circle.
Therefore, every point on AB, other than P, lies outside the circle.
This shows that AB meets the circle at point P.
Hence, AP is tangent to the circle at P.

Theorem 3: Prove that the lengths of tangents drawn from an external point to a circle are equal
Given: PT and PS are tangents from an external point P to the circle with centre O.
To prove: PT = PS
Construction: Join O to P, T and S.

Proof: In ∆OTP and ∆OSP.
OT = OS …[radii of the same circle]
OP = OP …[common]
∠OTP = ∠OSP …[each 90°]
∆OTP = ∆OSP …[R.H.S.]
PT = PS …[c.p.c.t.]

Note: If two tangents are drawn to a circle from an external point, then:

• They subtend equal angles at the centre i.e., ∠1 = ∠2.
• They are equally inclined to the segment joining the centre to that point i.e., ∠3 = ∠4.
∠OAP = ∠OAQ

The document Important Formulas: Circles | Mathematics (Maths) Class 10 is a part of the Class 10 Course Mathematics (Maths) Class 10.
All you need of Class 10 at this link: Class 10

Mathematics (Maths) Class 10

116 videos|420 docs|77 tests

FAQs on Important Formulas: Circles - Mathematics (Maths) Class 10

 1. What are the important formulas for finding the perimeter and area of a circle?
Ans. The important formulas for finding the perimeter and area of a circle are: (a) Perimeter Formula: P = 2πr, where P is the perimeter and r is the radius of the circle. (b) Area Formula: A = πr^2, where A is the area and r is the radius of the circle.
 2. What are the properties of a circle?
Ans. The properties of a circle are: (a) All points on the circumference of a circle are equidistant from the center. (b) The diameter of a circle is the longest chord that can be drawn in the circle. (c) The radius of a circle is a line segment connecting the center to any point on the circumference. (d) The circumference of a circle is the distance around the circle and is found using the formula C = 2πr, where C is the circumference and r is the radius. (e) The area of a circle is the amount of space enclosed by the circle and is found using the formula A = πr^2, where A is the area and r is the radius.
 3. How do you find the perimeter of a circle?
Ans. To find the perimeter of a circle, you can use the formula P = 2πr, where P is the perimeter and r is the radius of the circle. Multiply 2π by the radius of the circle to get the perimeter.
 4. How do you find the area of a circle?
Ans. To find the area of a circle, you can use the formula A = πr^2, where A is the area and r is the radius of the circle. Square the radius and multiply it by π to get the area.
 5. What is the relationship between the diameter and radius of a circle?
Ans. The diameter of a circle is twice the length of its radius. In other words, if you divide the diameter by 2, you will get the radius. Similarly, if you multiply the radius by 2, you will get the diameter.

Mathematics (Maths) Class 10

116 videos|420 docs|77 tests

Up next

 Explore Courses for Class 10 exam

Top Courses for Class 10

Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
10M+ students study on EduRev
Related Searches

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

;