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Chapter Analysis
Intermediate10 pages • EnglishQuick Summary
This chapter introduces the concept of tangents in relation to circles. It explores different cases for the tangents and secants and provides various geometric configurations, such as the angle between tangents and radii. The properties of tangents, including their lengths from external points, are explained using theorems and illustrated with examples. The chapter also provides mathematical proofs supporting these properties.
Key Topics
- •Tangent to a circle
- •Properties of tangents
- •Theorems involving tangents
- •Secant and tangent relationship
- •Angle between radius and tangent
- •External tangents
Learning Objectives
- ✓Understand the definition and properties of a tangent to a circle
- ✓Provide proofs for tangent theorems
- ✓Calculate the lengths of tangents from external points
- ✓Analyze geometric figures involving tangents and circles
- ✓Apply tangent properties to solve related mathematical problems
Questions in Chapter
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. The radius of the circle is (A) 7 cm (B) 12 cm (C) 15 cm (D) 24.5 cm
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In Fig. 10.11, if TP and TQ are the two tangents to a circle with centre O so that ∠POQ = 110°, then ∠PTQ is equal to (A) 60° (B) 70° (C) 80° (D) 90°
Page 152
If tangents PA and PB from a point P to a circle with centre O are inclined to each other at an angle of 80°, then ∠POA is equal to (A) 50° (B) 60° (C) 70° (D) 80°
Page 152
Prove that the tangents drawn at the ends of a diameter of a circle are parallel.
Page 152
Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre.
Page 152
The length of a tangent from a point A at distance 5 cm from the centre of the circle is 4 cm. Find the radius of the circle.
Page 153
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.
Page 153
A quadrilateral ABCD is drawn to circumscribe a circle. Prove that AB + CD = AD + BC.
Page 153
Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line-segment joining the points of contact at the centre.
Page 153
Additional Practice Questions
Describe the properties of a tangent line to a circle in relation to the radius at the point of contact.
mediumAnswer: A tangent line to a circle is perpendicular to the radius at the point of contact, meaning the angle between the tangent and the radius is 90 degrees.
What is the relationship between two tangents drawn from an external point to a circle?
easyAnswer: The lengths of two tangents drawn from an external point to a circle are equal.
Prove that the sum of angles in an isosceles triangle formed by a tangent and the radii of a circle is always 180 degrees.
mediumAnswer: In an isosceles triangle formed by a tangent and two radii, the base angles are equal, and the sum of all angles in the triangle must be 180 degrees as per the angle sum property.
How many tangents can be drawn from a single external point to a circle?
easyAnswer: Exactly two tangents can be drawn from a single external point to a circle.
Calculate the length of the tangent from a point 10 cm away from the center of a circle with a radius of 6 cm.
mediumAnswer: Using the Pythagorean theorem: length of tangent = √(10² - 6²) = 8 cm.