0 Comments
Class 9 Math, Euclid's Geometry Dear Class 9 students and fellow educators, 'Jsunil Sir' welcomes you to an enlightening journey into the world of Euclid's Geometry at JSUNIL TUTORIAL. This comprehensive guide offers a clear and easy-to-understand introduction to the principles laid down by the ancient mathematician Euclid. Whether you're a budding learner or a dedicated teacher, this resource is designed to deepen your understanding of geometry. Explore the foundations, sharpen your math skills, and lay a strong groundwork for academic success. Dive into Euclid's Geometry with us today EUCLID’S Postulates
1. A straight line may be drawn from any point to any other point. 2. A terminated line (line segment) can be produced indefinitely. 3. A circle may be described with any centre and any radius. 4. All right angles are equal to one another. 5. If a straight line falling on two straight lines makes the interior angles on the same side of it, taken together less than two right angles, then the the two straight lines if produced indefinitely, meet on that side on which the sum of angles is taken together less than two right angles. Euclid used the term postulate for the assumptions that were specific to geometry and otherwise called axioms. A theorem is a mathematical statement whose truth has been logically established. Visit Page Link to download solved Questions and Guess test Paper CBSE Class IX Introduction to Euclid's geometry Proof of this factor theorem
Let p(x) be a polynomial of degree greater than or equal to one and a be areal number such that p(a) = 0. Then, we have to show that (x – a) is a factor of p(x).Let q(x) be the quotient when p(x) is divided by (x – a). By remainder theorem Dividend = Divisor x Quotient + Remainder p(x) = (x – a) x q(x) + p(a) [Remainder theorem] ⇒ p(x) = (x – a) x q(x) [p(a) = 0]⇒ (x – a) is a factor of p(x) Conversely, let (x – a) be a factor of p(x). Then we have to prove that p(a) = 0 Now, (x – a) is a factor of p(x)⇒ p(x), when divided by (x – a) gives remainder zero. But, by the remainder theorem, p(x) when divided by (x – a) gives the remainder equal to p(a). ∴ p(a) = 0 Proof of remainder theorem. Let q(x) be the quotient and r(x) be the remainder obtained when the polynomial p(x) is divided by (x–a). Then, p(x) = (x–a) q(x) + r(x), where r(x) = 0 or some constant. Let r(x) = c, where c is some constant. Then p (x) = (x–a) q(x) + c Putting x = a in p(x) = (x–a) q(x) + c, we getp(a) = (a–a) q(a) + c ⇒ p(a) = 0 x q(a) + c ⇒ p(a) = c This shows that the remainder is p(a) when p(x) is divided by (x–a).1. [a is –8.] Download study material for polynomial Thales is the first mathematician credited with giving the first known proof “a circle is bisected by its diameter. One of Thales’ most famous pupils was Pythagoras and his group discovered many geometric properties and developed the theory of geometry to a great extent. At that time Euclid, a teacher of mathematics at Alexandria in Egypt, collected all the known work and arranged it in his famous treatise, called ‘Elements’. He divided the ‘Elements’ into thirteen chapters, each called a book. Some definitions from book -1 of Elements are: 1. A point is that which has no part. 2. A line is breathless length. 3. The ends of a line are points. 4. A straight line is a line which lies evenly with the points on itself. 5. A surface is that which has length and breadth only. 6. The edges of a surface are lines. 7. A plane surface is a surface which lies evenly with the straight lines on itself. Þ An axiom or a postulate is a mathematical statement which is assumed to be true without proof. These assumptions are actually obvious universal truths. Þ Theorems are statements which are proved, using definitions, axioms, previously proved statements and deductive reasoning. Þ Some of the Euclid’s axioms are: (i) Things which are equal to same thing are equal to one another. (ii) If equals are added to equals, the wholes are equal. (iii) If equals are subtracted from equals, the remainders are equals. (iv) Things which coincide with one another are equal to one another. (v) The whole is greater than the part. (vi) Things which are double of the same thing are equal to one another. (vii) Things which are halves of the same thing are equal to one another. Þ Euclid’s five postulates are: (i) A straight line may be drawn from any point to any other point. (ii) A terminated line can be produced indefinitely. (iii) A circle can be drawn with any centre and any radius. (iv) All right angles are equal to one another. (v) If a straight line falling on two straight lines makes the interior angles on the same side of it taken together less than two right angles, then the two straight lines, if produced indefinitely, meet on that side on which the sum of angles is less than two right angles. A system of axioms is called consistent, if it is impossible to deduce from these axioms a statement that contradicts any axioms or previously proved statement. Euclid’s first postulate can also be stated as below: Given two distinct points, there is a unique line that passes through them. Two distinct lines cannot have more than one point in common. Playfair’s Axiom: For every line l and for every point P not lying on l, there exists a unique line m, passing through P and parallel to l. [~ 5th Postulate] Two distinct intersecting lines cannot be parallel to the same line. CBSE Class IX Introduction to Euclid's geometry full study
9th Coordinate Geometry: Key concepts
ÞCoordinate Geometry: The branch of mathematics in which geometric problems are solved through algebra by using the coordinate system is known as coordinate geometry. Þ Coordinate axes: The position of a point in a plane is determined with reference to two fixed mutually perpendicular lines, called the coordinate axes. Þ Coordinate System, position of a point is described by ordered pair of two numbers. Þ Ordered pair: A pair of numbers a and b listed in a specific order with 'a' at the first place and 'b' at the second place is called an ordered pair (a, b). Note that (a, b) ¹ (b , a) and (x, y) = (y, x), if x = y. Þ P(a,b) be any point in the plane. 'a' the first number denotes the distance of point from Y-axis and 'b' the second number denotes the distance of point from X-axis. Þ The coordinates of origin are (0,0) Þ Every point on the x-axis is at a distance o unit from the X-axis. So its ordinate is 0. Þ Every point on the y-axis is at a distance of 0 unit from the Y-axis. So, its abscissa is 0. Þ The coordinates of a point on the x-axis are of the form (x, 0) and that of a point on the y-axis are (0, y). Þ A point in the first quadrant will be of the form (+, +). Similarly, a point in the second, third and fourth quadrants will be of the form (–, +), (–, –) and (+, –) respectively. Download links IX Co -Ordinate geometry Test Paper-1 IX Co -Ordinate geometry Test Paper-2 IX Co -Ordinate geometry Test Paper-3 IX Co -Ordinate geometry Test Paper-4 Assignment- IX Co -Ordinate geometry -5 Download above File |
Blog SeaRCH Link
All
Join Us For Update |