Exercise 10.1 Practice
Tangents to a Circle: Basic Concepts
Q1: Conceptual
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What is the maximum number of tangents that a single circle can have in total?
A tangent touches the circle at exactly one point.
Since a circle is made up of infinite points, we can draw a tangent at every single point on the circle.
A circle can have infinitely many tangents.
Q2: Fill in the Blanks
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Fill in the blanks:
(i) A tangent to a circle intersects it in _______ point(s).
(ii) A line intersecting a circle in two points is called a _______.
(iii) A circle can have _______ parallel tangents at the most.
(iv) The common point of a tangent to a circle and the circle is called _______.
(i) A tangent to a circle intersects it in _______ point(s).
(ii) A line intersecting a circle in two points is called a _______.
(iii) A circle can have _______ parallel tangents at the most.
(iv) The common point of a tangent to a circle and the circle is called _______.
(i) By definition, a tangent touches the circle at exactly one point.
(ii) A line that cuts through the circle at two distinct points is called a secant.
(iii) Parallel tangents can only be drawn at the endpoints of a diameter. Hence, there can be at most two parallel tangents.
(iv) The specific point where the tangent touches the circle is known as the point of contact.
Q3: Calculation (MCQ Style)
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A tangent PQ at a point P of a circle of radius 8 cm meets a line through the centre O at a point Q so that OQ = 17 cm. Find the length PQ.
Concept: The tangent at any point of a circle is perpendicular to the radius through the point of contact.
Therefore, $\angle OPQ = 90^\circ$. We have a right-angled triangle $\Delta OPQ$.
Given: Radius $OP = 8$ cm, Distance $OQ = 17$ cm.
Using Pythagoras Theorem:
$OQ^2 = OP^2 + PQ^2$
$17^2 = 8^2 + PQ^2$
$289 = 64 + PQ^2$
$OQ^2 = OP^2 + PQ^2$
$17^2 = 8^2 + PQ^2$
$289 = 64 + PQ^2$
$PQ^2 = 289 - 64 = 225$
$PQ = \sqrt{225} = 15$ cm.
$PQ = \sqrt{225} = 15$ cm.
Length PQ is 15 cm.
Q4: Construction Concept
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Draw a circle and two lines parallel to a given line such that one is a tangent and the other, a secant to the circle.
Explanation:
1. Draw the circle with any radius and a straight line outside or inside it (labeled 'Given Line').
2. To draw the Secant: Draw a line parallel to the given line that cuts through the circle at two distinct points.
3. To draw the Tangent: Draw a line parallel to the given line that touches the circle at exactly one point (either at the top or bottom of the circle relative to the given line).
See the diagram above for the visual representation.