# Math Blog

### Worksheet on Application Problems on Expansion of Powers of Binomials

Practice the questions given in the worksheet on application problems on expansion of powers of binomials and trinomials. 1. Use (a ± b)^2 = a^2 ± 2ab + b2 to evaluate the following: (i) (3.001)^2 (ii) (5.99)^2 (iii) 1001 × 999 (iv) 5.63 × 5.63 + 11.26 × 2.37 + 2.37 × 2.37

Continue reading "Worksheet on Application Problems on Expansion of Powers of Binomials"

### Application Problems on Expansion of Powers of Binomials & Trinomials

Here we will solve different types of application problems on expansion of powers of binomials and trinomials. 1. Use (x ± y)^2 = x^2 ± 2xy + y^2 to evaluate (2.05)^2. Solution: (2.05)^2 = (2 + 0.05)^2 = 2^2 + 2 × 2 × 0.05 + (0.05)^2 = 4 + 0.20 + 0.0025 = 4.2025.

Continue reading "Application Problems on Expansion of Powers of Binomials & Trinomials"

### Expansion of (x + a)(x + b)(x + c) | Solved Examples | Problems | Hint

We will discuss here about the expansion of (x + a)(x + b)(x + c). (x + a)(x + b)(x + c) = (x + a){(x + b)(x + c)} = (x + a){x$$^{2}$$ + (b + c)x + bc} = x{x$$^{2}$$ + (b + c)x + bc} + a{x$$^{2}$$ + (b + c)x + bc} = x$$^{3}$$ + (b + c)x$$^{2}$$ + bcx + ax$$^{2}$$ + a(b + c)x

Continue reading "Expansion of (x + a)(x + b)(x + c) | Solved Examples | Problems | Hint"

### Simplification of (a + b + c)(a$$^{2}$$+b$$^{2}$$+c$$^{2}$$–ab–bc– ca)

We will discuss here about the expansion of (a + b + c)(a$$^{2}$$ + b$$^{2}$$ + c$$^{2}$$ – ab – bc – ca). (a + b + c)(a$$^{2}$$ + b$$^{2}$$ + c$$^{2}$$ – ab – bc – ca) = a(a$$^{2}$$ + b$$^{2}$$ + c$$^{2}$$ – ab – bc – ca) + b(a$$^{2}$$ + b$$^{2}$$ + c$$^{2}$$ –ab – bc – ca)

Continue reading "Simplification of (a + b + c)(a$$^{2}$$+b$$^{2}$$+c$$^{2}$$–ab–bc– ca)"

### Simplification of (a ± b)(a^2 ∓ ab + b^2) | Sum or Difference of Cubes

We will discuss here about the expansion of (a ± b)(a$$^{2}$$ ∓ ab + b$$^{2}$$). (a + b)(a$$^{2}$$ - ab + b$$^{2}$$) = a(a$$^{2}$$ - ab + b$$^{2}$$) + b(a$$^{2}$$ - ab + b$$^{2}$$) = a$$^{3}$$ - a$$^{2}$$b + ab$$^{2}$$ + ba$$^{2}$$ - ab$$^{2}$$ + b$$^{3}$$ =

Continue reading "Simplification of (a ± b)(a^2 ∓ ab + b^2) | Sum or Difference of Cubes"

### Expansion of (a ± b)$$^{3}$$ | Algebraic Expressions and Formulas

We will discuss here about the expansion of (a ± b)$$^{3}$$. (a + b)$$^{3}$$ = (a + b) ∙ (a + b)$$^{2}$$ = (a + b)(a$$^{2}$$ + 2ab + b$$^{2}$$) = a(a$$^{2}$$ + 2ab + b$$^{2}$$) + b(a$$^{2}$$ + 2ab + b$$^{2}$$)=a$$^{3}$$+2a$$^{2}$$b+ab$$^{2}$$+ba$$^{2}$$+2ab$$^{2}$$+b$$^{3}$$

Continue reading "Expansion of (a ± b)$$^{3}$$ | Algebraic Expressions and Formulas"

### Express a^2 + b^2 + c^2 – ab – bc – ca as Sum of Squares

Here we will express a^2 + b^2 + c^2 – ab – bc – ca as sum of squares. If a, b, c are real numbers then (a – b)^2, (b – c)^2 and (c – a)^2 are positive as square of every real number is positive. So, a^2 + b^2 + c^2 – ab – bc – ca is always positive.

Continue reading "Express a^2 + b^2 + c^2 – ab – bc – ca as Sum of Squares"

### Worksheet on Simplification of (a + b)(a – b) | Hint | Answers

Practice the questions given in the worksheet on simplification of (a + b)(a – b). 1. Simplify by applying standard formula. (i) (5x – 9)(5x + 9) (ii) (2x + 3y)(2x – 3y) (iii) (a + b – c)(a – b + c) (iv) (x + y – 3)(x + y + 3) (v) (1 + a)(1 – a)(1 + a^2)

Continue reading "Worksheet on Simplification of (a + b)(a – b) | Hint | Answers"

### Worksheet on Completing Square |Find the Missing Term| Perfect Squares

Practice the questions given in the worksheet on completing square. Write the following as a perfect square. (i) 4X^2 + 4X + 1 (ii) 9a^2 – 12ab + 4b^2 (iii) 1 + 6/a + 9/a^2 2. Indicate the perfect squares among the following. Express each of the perfect squares as the square

Continue reading "Worksheet on Completing Square |Find the Missing Term| Perfect Squares"

### Completing a Square | Solved Examples on Completing a Square

Here we will learn how to completing a square.Problems on completing a square 1. What should be added to the polynomial 4m^2 + 8m so that it becomes perfect square? Solution: 4m^2 + 8m = (2m)^2 + 2 ∙ (2m) ∙ 2

Continue reading "Completing a Square | Solved Examples on Completing a Square"

### Worksheet on Expansion of (x ± a)(x ± b) | Find the Product | Answers

Practice the questions given in the worksheet on expansion of (x ± a)(x ± b). 1. (i) Find the product using standard formula. (i) (x + 2)(x + 5) (ii) (a – 4)(a – 7) (iii) (x + 1)(x – 8) (iv) (a – 3)(a + 2) (v) (3x + 1)(3x + 2) (vi) (4x – y)(4x + 2y) 2. Find the product.

Continue reading "Worksheet on Expansion of (x ± a)(x ± b) | Find the Product | Answers"

### Worksheet on Expanding of (a ± b ± c)^2 and its Corollaries | Answers

Practice the questions given in the worksheet on expanding of (a ± b ± c)^2 and its corollaries. 1. Expand the squares of the following trinomials. (i) a + 2b + 3c (ii) 2x + 3y + 4z (iii) x + 2y – 3z (iv) 3a – 4b – c (v) 1 – x - $$\frac{1}{x}$$ (vi) 1 – a – a^2. 2. Simplify:

Continue reading "Worksheet on Expanding of (a ± b ± c)^2 and its Corollaries | Answers"

### Worksheet on Expansion of (a ± b)^2 and its Corollaries | Answers

Practice the questions given in the worksheet on expansion of (a ± b)^2 and its corollaries. 1. Expand the squares of the following: (i) 4x + y (ii) 5a + 3b (iii) 2x + $$\frac{1}{x}$$ 2. Expand the following: (i) (x – 2y)^2 (ii) (3y – 2z)^2 (iii) (3x - $$\frac{1}{3x}$$)^2

Continue reading "Worksheet on Expansion of (a ± b)^2 and its Corollaries | Answers"

### Simplification of (a + b)(a – b) | | Solved Examples on Simplification

We will discuss here about the Simplification of (a + b)(a – b). (a + b)(a – b) = a(a – b) + b(a – b) = a$$^{2}$$ - ab + ba - b$$^{2}$$ = a$$^{2}$$ - b$$^{2}$$ Thus, we have (a + b)(a - b) = a$$^{2}$$ - b$$^{2}$$ Solved Examples on Simplification of (a + b)(a – b) 1.

Continue reading "Simplification of (a + b)(a – b) | | Solved Examples on Simplification"

### Expansion of (x ± a)(x ± b) | Special Identities | Expanding Binomials

We will discuss here about the expansion of (x ± a)(x ± b) (x + a)(x + b) = x(x + b) + a (x + b) = x^2 + xb + ax + ab = x^2 + (b + a)x + ab (x - a)(x - b) = x(x - b) - a (x - b) = x^2 - xb - ax + ab = x^2 - (b + a)x + ab (x + a)(x - b) = x(x - b) + a (x - b) = x^2 - xb

Continue reading "Expansion of (x ± a)(x ± b) | Special Identities | Expanding Binomials"

### Expansion of (a ± b ± c)^2 | Square of a Trinomial | Algebra Formulas

We will discuss here about the expansion of (a ± b ± c)^2. (a + b + c)^2 = {a + (b + c)}^2 = a^2 + 2a(b + c) + (b + c)^2 = a^2 + 2ab + 2ac + b^2 + 2bc + c^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca) = sum of squares of a, b, c + 2(sum of the products of a, b, c taking two at a ti

Continue reading "Expansion of (a ± b ± c)^2 | Square of a Trinomial | Algebra Formulas"

### A Rhombus is a Parallelogram whose Diagonals Meet at Right Angles

Here we will prove that a rhombus is a parallelogram whose diagonals meet at right angles. Given: PQRS is a rhombus. So, by definition, PQ = QR = RD = SP. Its diagonals PR and QS intersect at O. To prove: (i) PQRS is a parallelogram. (ii) ∠POQ = ∠QOR = ∠ROS = ∠SOP = 90°.

Continue reading "A Rhombus is a Parallelogram whose Diagonals Meet at Right Angles"

### Pair of Opposite Sides of a Parallelogram are Equal and Parallel

Here we will discuss about one of the important geometrical property of parallelogram. A quadrilateral is a parallelogram if one pair of opposite sides are equal and parallel Given: PQRS is a quadrilateral in which PQ = SR and PQ ∥ SR. To prove: PQRS is a parallelogram.

Continue reading "Pair of Opposite Sides of a Parallelogram are Equal and Parallel"

### A Quadrilateral is a Parallelogram if its Diagonals Bisect each Other

Here we will discuss about a quadrilateral is a parallelogram if its diagonals bisect each other. Given: PQRS is a quadrilateral whose diagonals PR and QS bisect each other at O, i.e., OP = OR and OQ = OS. To prove: PQRS is a parallelogram. Proof: In ∆OPQ and ∆ORS, OP = OR

Continue reading "A Quadrilateral is a Parallelogram if its Diagonals Bisect each Other"

### Diagonals of a Parallelogram Bisect each Other | Diagonals Bisect each

Here we will discuss about the diagonals of a parallelogram bisect each other. In a parallelogram, diagonals bisect each other and each diagonal bisects the parallelogram into two congruent triangles. Given: PQRS is a parallelogram in which PQ ∥ SR and PS ∥ QR. Its diagonals

Continue reading "Diagonals of a Parallelogram Bisect each Other | Diagonals Bisect each"

### Expansion of (a ± b)^2 | Power of the Trinomial | Algebraic Expression

A binomial is an algebraic expression which has exactly two terms, for example, a ± b. Its power is indicated by a superscript. For example, (a ± b)2 is a power of the binomial a ± b, the index being 2. A trinomial is an algebraic expression which has exactly three terms

Continue reading "Expansion of (a ± b)^2 | Power of the Trinomial | Algebraic Expression"

### Opposite Angles of a Parallelogram are Equal | Related Solved Examples

Here we will discuss about the opposite angles of a parallelogram are equal. In a parallelogram, each pair of opposite angles are equal. Given: PQRS is a parallelogram in which PQ ∥ SR and QR ∥ PS To prove: ∠P = ∠R and ∠Q = ∠S Construction: Join PR and QS. Proof: Statement:

Continue reading "Opposite Angles of a Parallelogram are Equal | Related Solved Examples"

### Opposite Sides of a Parallelogram are Equal | Solved Examples

Here we will discuss about the opposite sides of a parallelogram are equal in length. In a parallelogram, each pair of opposite sides are of equal length. Given: PQRS is a parallelogram in which PQ ∥ SR and QR ∥ PS. To prove: PQ = SR and PS = QR. Construction: Join PR

Continue reading "Opposite Sides of a Parallelogram are Equal | Solved Examples"

### Concept of Parallelogram |Quadrilateral| Rectangle| Rhombus| Trapezium

Here we will discuss about the concept of parallelogram. Quadrilateral: A rectilinear figure enclosed by four line segments is called a quadrilateral. In the adjoining figures, we have two quadrilaterals PQRS, each enclosed by four line segments PQ, QR, RS and SP which

### Diagonal of a Quadrilateral Divides it in Two Triangles of Equal Area

Here we will prove that if each diagonal of a quadrilateral divides it in two triangles of equal area then prove that the quadrilateral is a parallelogram. Solution: Given: PQRS is a quadrilateral whose diagonals PR and QS cut at O such that ar(∆PQR) = ar(∆PSR), and

Continue reading "Diagonal of a Quadrilateral Divides it in Two Triangles of Equal Area"

### The Area of a Rhombus is Equal to Half the Product of its Diagonals

Here we will prove that the area of a rhombus is equal to half the product of its diagonals. Solution: Given: PQRS is a rhombus whose diagonals are PR and QS. The diagonals intersect at O. To prove: ar(rhombus PQRS) = 1/2 ×PR × QS. Statement ar(∆RSQ) = 1/2 ×Base × Altitude

Continue reading "The Area of a Rhombus is Equal to Half the Product of its Diagonals"

### Area of the Triangle formed by Joining the Middle Points of the Sides

Here we will prove that the area of the triangle formed by joining the middle points of the sides of a triangle is equal to one-fourth area of the given triangle. Solution: Given: X, Y and Z are the middle points of sides QR, RP and PQ respectively of the triangle PQR.

Continue reading "Area of the Triangle formed by Joining the Middle Points of the Sides"

### Problems on Finding Area of Triangle and Parallelogram | With Diagram

Here we will learn how to solve different types of problems on finding area of triangle and parallelogram. 1. In the figure, XQ ∥ SY, PS ∥ QR, XS ⊥ SY, QY ⊥ SY and QY = 3 cm. Find the areas of ∆MSR and parallelogram PQRS. Solution: ar(∆MSR) = 1/2 × ar(rectangle of SR

Continue reading "Problems on Finding Area of Triangle and Parallelogram | With Diagram"

### Triangles with Equal Areas on the Same Base have Equal Corresponding..

Here we will prove that triangles with equal areas on the same base have equal corresponding altitudes (or are between the same parallels). Given: PQR and SQR are two triangles on the same base QR, and ar(∆PQR) = ar(∆SQC). Also, PN and SM are their corresponding altitudes.

Continue reading "Triangles with Equal Areas on the Same Base have Equal Corresponding.."

### Triangles on the Same Base & between Same Parallels are Equal in Area

Here we will prove that triangles on the same base and between the same parallels are equal in area. Given: PQR and SQR are two triangles on the same base QR and are between the same parallel lines QR and MN, i.e., P and S are on MN. To prove: ar(∆PQR) = ar(∆SQR)

Continue reading "Triangles on the Same Base & between Same Parallels are Equal in Area"

### Area of a Triangle is Half that of a Parallelogram on the Same Base

Here we will prove that the area of a triangle is half that of a parallelogram on the same base and between the same parallels. Given: PQRS is a parallelogram and PQM is a triangle with the same base PQ, and are between the same parallel lines PQ and SR.

Continue reading "Area of a Triangle is Half that of a Parallelogram on the Same Base"

### Area of a Parallelogram is Equal to that of a Rectangle Between ......

Here we will prove that the area of a parallelogram is equal to that of a rectangle on the same base and of the same altitude, that is between the same parallel lines. Given: PQRS is a parallelogram and PQ MN is a rectangle on the same base PQ and between the same parallel

Continue reading "Area of a Parallelogram is Equal to that of a Rectangle Between ......"

### Parallelogram on the Same Base and Between the Same Parallel Lines

Here we will prove that parallelogram on the same base and between the same parallel lines are equal in area. Given: PQRS and PQMN are two parallelograms on the same base PQ and between same parallel lines PQ and SM. To prove: ar(parallelogram PQRS) = ar(parallelogram PQMN).

Continue reading "Parallelogram on the Same Base and Between the Same Parallel Lines"

### Diagonal of a Parallelogram Divides it into Two Triangles of EqualArea

Here we will prove that every diagonal of a parallelogram divides it into two triangles of equal area. Given: PQRS is a parallelogram in which PQ ∥ SR and SP ∥ RQ. PR is a diagonal of the parallelogram. To prove: ar(∆PSR) = ar(∆RQP). Proof: Statement 1. ∠SPR = ∠PRQ. 2. ∠SRP

Continue reading "Diagonal of a Parallelogram Divides it into Two Triangles of EqualArea"

### Base and Height (Altitude) in a Triangle and a Parallelogram | Diagram

We will discuss here about the Base and height (altitude) in a triangle and a parallelogram. In ∆PQR, any side may be taken as the base. If QR is taken as the base then the perpendicular PM on QR is the corresponding altitude (height) of the triangle. In the parallelogram

Continue reading "Base and Height (Altitude) in a Triangle and a Parallelogram | Diagram"

### Area of a Closed Figure |Measurement of Area |Area Axiom for Rectangle

We will discuss here about the area of a closed figure, measurement of area, area axiom for rectangle, area axiom for congruent figures and addition axiom for area. The measure of the reason bounded by a closed figure in a plane is called its area. In the following the areas

Continue reading "Area of a Closed Figure |Measurement of Area |Area Axiom for Rectangle"

### Bisectors of the Angles of a Parallelogram form a Rectangle | Diagram

Here we will prove that the bisectors of the angles of a parallelogram form a rectangle. Given: PQRS is a parallelogram in which PQ ∥ SR and SP ∥ RQ. The bisectors of ∠P, ∠Q, ∠R and ∠S are PJ, QK, RL and SM respectively which enclose the quadrilateral JKLM. To prove: JKLM is

Continue reading "Bisectors of the Angles of a Parallelogram form a Rectangle | Diagram"

### Conditions for Classification of Quadrilaterals and Parallelograms

We will discuss here about Conditions for classification of quadrilaterals and parallelograms. On the basis of the above definitions, theorems and converse propositions we conclude the following. 1. A quadrilateral is a parallelogram if any one of the following holds.

### Diagonals of a Parallelogram are Equal & Intersect at Right Angles

Here we will prove that if in a parallelogram the diagonals are equal in length and intersect at right angles, the parallelogram will be a square. Given: PQRS is a parallelogram in which PQ ∥ SR, PS ∥ QR and diagonal PR ⊥diagonal QS. To prove: PQRS is a square, i.e., PQ

Continue reading "Diagonals of a Parallelogram are Equal & Intersect at Right Angles"

### Diagonals of a Square are Equal in Length & they Meet at Right Angles

Here we will prove that in a square, the diagonals are equal in length and they meet at right angles. Given: PQRS is a square in which PQ = QR = RS = SP, and ∠QPS = ∠PQR = ∠QRS = ∠RSP = 90°. To prove: PR = QS and PR ⊥ QS Proof: Statement 1. In ∆SPQ and ∆RQP, (i) SP = QR

Continue reading "Diagonals of a Square are Equal in Length & they Meet at Right Angles"

### A Parallelogram, whose Diagonals are of Equal Length, is a Rectangle

Here we will prove that a parallelogram, whose diagonals are of equal length, is a rectangle. Given: PQRS is a parallelogram in which PQ ∥ SR, PS ∥ QR and PR = QS. To prove: PQRS is a parallelogram, i.e., in the parallelogram PQRS, one angle, say ∠QPS = 90°. Proof: In ∆PQR

Continue reading "A Parallelogram, whose Diagonals are of Equal Length, is a Rectangle"

### In a Rectangle the Diagonals are of Equal Lengths | Proof | Diagram

Here we will prove that in a rectangle the diagonals are of equal lengths. Given: PQRS is rectangle in which PQ ∥ SR, PS ∥ QR and ∠PQR = ∠QRP = ∠RSP = ∠SPQ = 90°. To prove: The diagonals PR and QS are equal. Proof: Statement In ∆PQR and ∆RSP 1.∠QPR = ∠SRP 2. ∠QRP = ∠SPR

Continue reading "In a Rectangle the Diagonals are of Equal Lengths | Proof | Diagram"

### A Parallelogram whose Diagonals Intersect at Right Angles is a Rhombus

Here we will prove that a parallelogram, whose diagonals intersect at right angles, is a rhombus. Given: PQRS is a parallelogram in which PQ ∥ SR, PS ∥ QR and ∠QOR = ∠POQ = ∠ROS = ∠POS = 90°. To prove: PQRS is a rhombus, i.e., PQ = QR = RS = SP. Proof: In ∆PQR and ∆RSP,

Continue reading "A Parallelogram whose Diagonals Intersect at Right Angles is a Rhombus"

### What is Rectilinear Figure? | What is Diagonal of a Polygon? | Polygon

What is rectilinear figure? A plane figure whose boundaries are line segments is called a rectilinear figure. A rectilinear figure may be closed or open. Polygon: A closed plane figures whose boundaries are line segments is called a polygon. The line segments are called its

Continue reading "What is Rectilinear Figure? | What is Diagonal of a Polygon? | Polygon"

### Sum of the Exterior Angles of an n-sided Polygon | Solved Examples

Here we will discuss the theorem of the sum of all exterior angles of an n-sided polygon and sum related example problems. 2. If the sides of a convex polygon are produced in the same order, the sum of all the exterior angles so formed is equal to four right angles.

Continue reading "Sum of the Exterior Angles of an n-sided Polygon | Solved Examples"

### Sum of the Interior Angles of an n-sided Polygon | Related Problems

Here we will discuss the theorem of sum of the interior angles of an n-sided polygon and some related example problems. The sum of the interior angles of a polygon of n sides is equal to (2n - 4) right angles. Given: Let PQRS .... Z be a polygon of n sides.

Continue reading "Sum of the Interior Angles of an n-sided Polygon | Related Problems"

### Area and Perimeter of Combined Figures | Circle | Triangle |Square

Here we will solve different types of problems on finding the area and perimeter of combined figures. 1. Find the area of the shaded region in which PQR is an equilateral triangle of side 7√3 cm. O is the centre of the circle. (Use π = $$\frac{22}{7}$$ and √3 = 1.732.)

Continue reading "Area and Perimeter of Combined Figures | Circle | Triangle |Square"

### Area of a Circular Ring | Radius of the Outer Circle and Inner Circle

Here we will discuss about the area of a circular ring along with some example problems. The area of a circular ring bounded by two concentric circle of radii R and r (R > r) = area of the bigger circle – area of the smaller circle = πR^2 - πr^2 = π(R^2 - r^2)

Continue reading "Area of a Circular Ring | Radius of the Outer Circle and Inner Circle"

### Area and Perimeter of a Semicircle | Solved Example Problems | Diagram

Here we will discuss about the area and perimeter of a semicircle with some example problems. Area of a semicircle = $$\frac{1}{2}$$ πr$$^{2}$$ Perimeter of a semicircle = (π + 2)r. Solved example problems on finding the area and perimeter of a semicircle

Continue reading "Area and Perimeter of a Semicircle | Solved Example Problems | Diagram"