Pythagoras Theorem
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Here is a detailed explanation of Pythagoras Theorem. This theorem is applicable to Right Angle Triangles only. Hope you will get a better understanding through this presentation. Thank You.. Regards Mukul Garg
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Pythagoras' theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. It introduces Pythagorean triples, where the sides of a right triangle have integer values that follow this relationship. The proof of the theorem shows that the areas of squares constructed on the sides of a right triangle follow the same relationship, demonstrating why the hypotenuse must be the longest side. The theorem is only applicable to right triangles.
Trignometry
Trigonometry is the study of relationships between the sides and angles of triangles. It originated from Greek terms meaning "three", "sides", and "measure". Trigonometry uses right triangles to define six trigonometric ratios relating the sides and angles. The Pythagorean theorem, which states that the square of the hypotenuse equals the sum of the squares of the other two sides, is fundamental to trigonometry. Trigonometric functions like sine, cosine, and tangent can be used to solve problems involving distances and angles.
Quadrilaterals
1) A quadrilateral is a geometric figure with four sides, four angles, and two diagonals. The sum of the angles is always 360 degrees.
2) There are six types of quadrilaterals: trapezoid, parallelogram, rectangle, rhombus, square, and kite. A parallelogram has both pairs of opposite sides parallel. A rectangle has one right angle. A square is both a rectangle and rhombus with all sides equal.
3) Theorems include: the diagonals of a parallelogram bisect each other; if the diagonals of a quadrilateral bisect each other it is a parallelogram; a quadrilateral is
CST 504 Distance in the Cartesian Plane
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Trigonometry class10
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2) Examples of right triangles are used to derive the trigonometric ratios of angles like sin(45°)=1/√2 and tan(300°)=√3.
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Shivam goyal ix e
The document discusses various properties of quadrilaterals:
- It defines the six types of quadrilaterals - trapezium, parallelogram, rectangle, rhombus, square, and kite.
- It provides examples and definitions for each type.
- Several theorems regarding the properties of parallelograms are presented, including that the diagonals of a parallelogram bisect each other and that opposite sides of a parallelogram are equal.
- Additional theorems state that a quadrilateral is a parallelogram if opposite sides are equal or if opposite angles are equal.
Trigonometry part 1
This document discusses trigonometry and trigonometric ratios. It defines trigonometry as the study of relationships between the sides and angles of a triangle. It then defines the trigonometric ratios of sine, cosine, and tangent for a right triangle in terms of the sides adjacent to, opposite to, and hypotenuse of an angle. It provides examples of using trigonometric ratios to solve for missing sides or angles of a right triangle. The document also discusses trigonometric ratios of complementary angles and provides the specific trigonometric ratios for 30, 60, 45, and 90 degree angles.
class-9-math-triangles_1595671835220.pdf
1. A triangle has three sides, three angles, and three vertices.
2. There are five criteria to determine if two triangles are congruent: side-angle-side, angle-side-angle, angle-angle-side, side-side-side, and right-angle-hypotenuse-side.
3. Properties of triangles include: angles opposite equal sides are equal; sides opposite equal angles are equal; the longer the side, the larger the opposite angle; and the sum of any two sides is greater than the third side.
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The document discusses various properties and theorems related to triangles. It begins by defining different types of triangles based on side lengths and angle measures. It then covers the four congruence rules for triangles: SAS, ASA, AAS, and SSS. The document proceeds to prove several theorems about relationships between sides and angles of triangles, such as opposite sides/angles of isosceles triangles being equal, larger sides having greater opposite angles, and the sum of any two angles being greater than the third side. It concludes by proving that the perpendicular from a point to a line is the shortest segment.
Congruence of Triangle
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- The SSS (Side-Side-Side) rule, which states two triangles are congruent if three sides of one triangle are equal to the corresponding three sides of the other triangle. An example proof is also provided.
- The Hypotenuse-Leg rule, which states two right triangles are congruent if the hypotenuse and one side of one
Similar to Pythagoras Theorem (20)
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Pythagoras Theorem
- 1. PYTHAGORAS THEOREM (PROOF OF PYTHAGORAS THEOREM)
- 2. STATEMENT :- According to the Pythagoras theorem “In a Right angled Triangle, the square of the hypotenuse is equal to the sum of individual squares of the other two sides”. i.e (Hypotenuse)^2 = (Base)^2 + (Perpendicular)^2
- 3. Right Angled Triangle hypotneuse base perpendicular A B C Acc. To Pythgoras Theorem (Hypotneuse)^2 = (Base)^2 + (Perpendicular)^2 (AC)^2 = (AB)^2 + (BC)^2
- 4. Proof of Pythagoras Theorem A B C D x 90 - x 90 90 ABC is a right angled triangle .Draw a perpendicular BD on AC. Now , there are two more right angle triangle i.e, tri.(BCD) and tri.(BAD) Therefore, AC = AD + CD In tri.(BCD), Cos x = base hypotneuse Cos x = CD BC So, CD= BC Cosx--eq1 In tri.(BAD), Cos (90 - x) = base hypotneuse Cos (90 - x) = AD AB So, AD = AB Sinx --eq2 (cos(90-x) = sinx) Also, in tri(ABC), Sin x = perpendicular hypotneuse Sinx = AB AC Cos x = base hypotneuse Cos x = BC AC
- 5. Since, AC = AD + CD By putting value of AD and CD from eq1 and eq2 We have, AC = AB Sin x + BC Cos x Now, putting Sin x = AB and Cos x = BC in above equation, AC AC We have, AC = AB( AB ) + BC( BC ) ( AC ) ( AC) AC x AC = (AB)^2 + (BC)^2 (AC)^2 = (AB)^2 + (BC)^2
- 6. THANK YOU