Unraveling the Mysteries of Circles: A Journey Through Geometry
Are you ready to explore the fascinating world of Circles? This essential chapter from the Class 10 NCERT syllabus introduces us to a fundamental concept seen all around us—from the wheels of vehicles to the planets in orbit. Understanding circles helps us unlock the deeper connections between geometry and the real world. In this blog, we'll delve into key concepts such as tangents, theorems, and properties, while uncovering how circles form the foundation of more advanced mathematical ideas. Let’s embark on this geometrical journey and discover the elegance and power of circles!
EXERCISE 10.1
1. How many tangents can a circle have?
Ans: A circle can have infinite tangents due to its unique properties.
1) From Points Outside the Circle: There are infinitely many pairs of tangents for each point outside the circle, as there are exactly two tangents drawn to the circle for each point.
2) From Points on the Circle: The circumference of a circle can be drawn from every point, resulting in infinitely many tangents, as there are infinitely many points on the circle's circumference.
2. 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 one point (s).(ii) A line intersecting a circle in two points is called a secant.(iii) A circle can have two parallel tangents at the most.(iv) The common point of a tangent to a circle and the circle is called the point
of contact.
3. A tangent PQ at a point P of a circle of radius 5 cm meets a line through the
center O at a point Q so that OQ = 12 cm. Length PQ is :
(A) 12 cm (B) 13 cm (C) 8.5 cm (D) √119 cm.
Solution:
1) PQ is the tangent to the circle with center O at point P.
2) OP = 5 cm is the radius, and OQ = 12 cm.
(PQ)2 + (OP)2 = (OQ)2
(PQ)2 = (OQ)2 – (OP)2
(PQ)2 = (12)2 – (5)2
(PQ)2 = 144 – 25
(PQ)2 = 119
PQ = √119
4) Ans: (D) PQ = √119 cm.
4. 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.
Solution:
As we conclude this journey into the world of Circles, we realize how this seemingly simple shape holds profound significance in geometry and the real world. From theorems about tangents to understanding radii and chords, circles provide us with valuable mathematical tools and insights. By mastering the concepts in this chapter, you're not just preparing for exams but also sharpening your problem-solving skills for life.
Remember, the study of circles is not just about formulas and theorems—it's about recognizing the patterns and harmony in the universe around us. Keep practicing, keep exploring, and let the beauty of mathematics guide you forward!
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