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Circles Notes: Class 10 Maths Chapter 10
Class 10 Maths Chapter 10 Circles covers tangents to a circle, the number of tangents from a point, and key theorems on tangent lines.
Overview of Topics & Subtopics (Click to Jump)
1. Tangent to a Circle
Short Note:
• A tangent to a circle is a line that touches the circle at exactly one point, called the point of contact.
• A tangent to a circle is perpendicular to the radius through the point of contact.
2. Number of Tangents from a Point on a Circle
Short Note:
• From a point on the circle, there is exactly one tangent.
• From a point inside the circle, no tangent can be drawn.
• From a point outside the circle, exactly two tangents can be drawn.
3. Length of Tangents from an External Point
Short Note:
• The lengths of the two tangents drawn from an external point to a circle are equal.
NCERT Textbook Exercise Questions
Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact.
Answer: Let O be the centre, P the point of contact, and XY the tangent. For any other point Q on XY, OQ > OP since P is the closest point on the tangent to O (OP is the radius, and any other point on the tangent line lies outside the circle). This means OP is the shortest distance from O to line XY, so OP ⊥ XY.
From a point Q, the length of the tangent to a circle is 24 cm and the distance of Q from the centre is 25 cm. Find the radius of the circle.
Answer: Since tangent ⊥ radius at the point of contact, by Pythagoras theorem: OQ² = OP² + PQ² ⇒ 25² = r² + 24² ⇒ r² = 625−576 = 49 ⇒ r = 7 cm.
Two concentric circles are of radii 5 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle.
Answer: The chord touches the smaller circle at its midpoint, forming a right angle with the radius (3 cm) at that point. Half-chord = √(5²−3²) = √16 = 4 cm, so full chord = 8 cm.
A quadrilateral ABCD is drawn to circumscribe a circle. Prove that AB + CD = AD + BC.
Answer: Using the property that tangents from an external point are equal in length, let the tangent lengths from A, B, C, D to the points of contact be a, b, c, d respectively (as appropriate pairs). Then AB = a+b, BC = b+c, CD = c+d, DA = d+a. Adding AB+CD = a+b+c+d = AD+BC, proving the result.
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