What is the Pythagorean Theorem The Pythagorean theorem is related to the study of sides of a right angled triangle. It is also called as pythagoras theorem. The pythagorean theorem states that, In a right triangle, (length of the hypotenuse)2 = {(1st side)2 + (2nd side)2}. In a right angled triangle, there are three sides: hypotenuse, perpendicular and base. The base and the perpendicular make an angle of 90 degree with eachother. So, according to pythagorean theorem: (Hypotenuse)2 = (Perpendicular)2 + (Base)2 Pythagorus Theorem Proof From the above figure 2, Δ ABC is a right angled triangle at angle C. From C put a perpendicular to AB at H. Know More About Frequency Table Worksheet
Now consider the two triangles Δ ABC and Δ ACH, these two triangles are similar to each other because of AA similarity. This is because both the triangle have a right angle and one common angle at A. So by these similarity, ac = ea and bc = db a2 = c*e and b2 = c*d Sum the a2 and b2, we get a2 + b2 = c*e + c*d a2 + b2 = c(e + d) a2 + b2 = c2 (since e + d = c) Hence Proved.
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Radius of a Circle Circle is defined as the set of points that is at an equal distant from the centre of the circle. There are a number of terminologies involved in a Circle. Some of them are as follows: Centre: The predetermined point from which the surface of the circle is at an equidistant is called the centre of a circle. Radius: The constant distance from the centre to a point on the surface of the circle is called its radius . Circumference: The boundary of a circle is called its circumference. Chord: A line segment whose end points is present on the circumference of a circle is called a chord . Diameter: A chord crossing through the midpoint of a circle is called its diameter.
Circle Formulas Diameter of a Circle: Diameter = 2 X Radius Radius of a Circle: Radius(R) = Diameter / 2 Area of a Circle: Area = pi X R2 Circumference of a Circle: Circumference = 2 X pi X R Circle Theorem Theorem 1: A perpendicular from the centre of a circle to a chord bisects the chord. Given : AB is a chord in a circle with centre O. OC ⊥ AB. To prove: The point C bisects the chord AB. Construction: Join OA and OB Proof: In triangles OAC and OBC, m∠OCA = m∠OCB = 90 (Given) Read More About Polynomials Worksheet
OA = OB (Radii) OC = OC (common side) ∠OAC = ∠OBC (RHS) CA = CB (corresponding sides) The point C bisects the chord AB. Hence the theorem is proved. Theorem 2: AB and CD are equal chords of a circle whose centre is O. OM ⊥ AB and ON ⊥ CD. Prove that m∠OMN = m∠ONM.
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