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Questions and Answers
What is the definition of a circle?
What is the formula for calculating the circumference of a circle?
Which term refers to a straight line segment that intersects the circle at two points?
What is the region enclosed by a circle called?
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Which formula can be used to generate a Pythagorean triple?
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What is the equation of a circle with center (0, 0) and radius 7?
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What is the area of a circle with radius 10?
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What is the equation of a circle with center (-4, 5) and radius 3?
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For a right-angled triangle, what is the relationship between the squares of the lengths of its sides?
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What is one Pythagorean triple that satisfies the equation $a^2 + b^2 = c^2$?
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Study Notes
Maths CBSE 10 CLASS: Understanding Circles
Maths is an essential subject, and understanding circles is a crucial aspect of the 10th-grade curriculum in the CBSE (Central Board of Secondary Education) syllabus. This article aims to provide a comprehensive understanding of circles, their properties, and various related concepts.
Circles and Their Properties
Definition of a Circle
A circle is the set of all points in a plane that are equidistant from a fixed point called the center. The fixed point is called the center, and the distance from the center to any point on the circle is called the radius. The longest distance from one point on the circle to another is called the diameter.
Radius and Diameter
The radius of a circle is the distance from its center to any point on the circle, and it is half the length of the diameter. The diameter is the longest distance from one point on the circle to another, passing through the center.
Circumference of a Circle
The circumference of a circle is the distance around it. It is calculated using the formula C = 2Ï€r, where r is the radius of the circle.
Area of a Circle
The area of a circle is the region enclosed by the circle. It is calculated using the formula A = πr^2, where r is the radius of the circle.
Chord, Arc, and Sector
A chord is a straight line segment that intersects the circle at two points. An arc is a part of the circle. A sector is the region bounded by two radii and an arc.
Important Theorems and Formulae
Pythagoras Theorem
The theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. Mathematically, this can be written as c^2 = a^2 + b^2, where c is the hypotenuse, and a and b are the other two sides.
Formulae for the Pythagorean Triple
A Pythagorean triple is a set of three integers a, b, and c such that a^2 + b^2 = c^2. The formulae for generating Pythagorean triples are as follows:
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a = 2mn, b = m^2 - n^2, c = m^2 + n^2, where m and n are integers with no common factors other than 1. -
a = 2ab, b = a^2 - c^2, c = a^2 + b^2, where a, b, and c are integers with no common factors other than 1.
Formulae for the Area of a Circle
The area of a circle can be calculated using the formula A = πr^2, where r is the radius of the circle.
Examples and Solved Problems
Example 1
Find the equation of the circle with center (2, 3) and radius 5.
Solution: The equation of a circle with center (h, k) and radius r is given by (x - h)^2 + (y - k)^2 = r^2. In this case, h = 2, k = 3, and r = 5, so the equation of the circle is (x - 2)^2 + (y - 3)^2 = 5^2.
Example 2
Find the equation of the circle with center (3, -4) and passing through the point (5, 2).
Solution: Let (x, y) be any point on the circle. Then, ((x - 3)^2 + (y + 4)^2 = 10^2). Since the point (5, 2) lies on the circle, we have (5 - 3)^2 + (2 + 4)^2 = 10^2, which simplifies to 16 + 16 = 100, or x = 10, y = 6. Therefore, the equation of the circle is (x - 3)^2 + (y + 4)^2 = 100.
Conclusion
Understanding circles, their properties, and related concepts is an essential part of the CBSE 10th-grade maths curriculum. By learning and practicing the concepts discussed in this article, students can gain a solid foundation in mathematics, which will serve them well in their future academic pursuits.
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Description
This quiz covers the fundamental concepts related to circles in the CBSE (Central Board of Secondary Education) 10th-grade syllabus, including definitions, properties, theorems, formulae, and solved problems. Topics include the definition of a circle, radius and diameter, circumference, area, chords, arcs, sectors, Pythagoras theorem, Pythagorean triples, and equations of circles.
