What Are The Properties Of CirclesTwo circles are congruent, if and only if they have equal radii. Two arcs of a circle are congruent if the angles subtended by them at the centre are equal. Two arcs subtend equal angles at the centre, if the arcs are c…

Basic Proportionality Theorem or Thales TheoremStatement: If a line is drawn parallel to one side of a triangle intersecting the other two sides, then it divides the two sides in the same ratio. Given: A triangle ABC in which DE || BC, and intersects AB…

Construction Of Perpendicular Bisector Of A Line SegmentA line which is perpendicular to a given line segment (AB) and divides it into two equal halves, i.e., AO = OB is called the perpendicular bisector of AB. In figure, XY is the perpendicular bisector…

Types Of Factorization Example Problems With SolutionsType I: Factorization by taking out the common factors.  Example 1:    Factorize the following expression 2x2y + 6xy2 + 10x2y2 Solution:    2x2y + 6xy2 + 10x2y2 =2xy(x + 3y + 5xy) Type II: Factorizati…

Construction Of The Bisector Of A Given AngleGiven: An angle CAB To construct: Bisector of ∠CAB.Step 1: Taking A as the centre and with any suitable radius, draw an arc cutting the arms AB and AC of ∠CAB at D and E respectively. Step 2: Taking D as the c…

Angle Sum Property of a TriangleTheorem 1: Prove that sum of all three angles is 180° or 2 right angles. Given: ∆ABC To prove: ∠A + ∠B + ∠C = 180° Construction: Draw PQ || BC, passes through point A. Proof: ∠1 = ∠B   and  ∠3 = ∠C         ……. (i) [∵ alt…

Factorization Of Algebraic Expressions Of The Form a3 + b3 + c3, When a + b + c = 0Example 1:   Factorize (x – y)3 + (y – z)3 + (z – x)3 Solution:    Let x – y = a, y– z = b and z – x = c, then, a + b + c = x – y + y – z + z –x = 0 ∴ a3 + b3 + c3 = 3abc …

What is the Heron’s formulaHeron’s formula: If we have all sides of triangle and their is no way to find height then we use this formula for area of triangle. Heron’s Formula Example Problems With SolutionsExample 1:    For given figure find the s (s – …

Equations Of Lines Parallel To The X-Axis And Y-AxisWe can represent graph of these equations in two types of geometrically (A) in one variable or on number line (B) in two variable or on the Cartesian plane In one variable, the solution is represent by…

Rounding and Error in MeasurementRounding as associated with Error: Error in measurement may also be indicated by the terms “rounded” or “to the nearest“. When dealing with measurements that have been rounded, remember what happens when rounding occurs.…