Algebra Play Class 8 Notes Maths Part 2 Chapter 6 - #NCSOLVE 📚

Students often refer to Class 8 Maths Notes and Part 2 Chapter 6 Algebra Play Class 8 Notes during last-minute revisions.

Class 8 Maths Chapter 6 Algebra Play Notes

Class 8 Algebra Play Notes

→ Algebra is very useful in modeling and understanding numerical scenarios. Because of this, it occurs in almost all areas of mathematics, science and beyond.

→ Algebra is an indispensable tool in justifying mathematical statements.

→ We applied algebra to analyse ‘Think of a Number’ tricks, number pyramids, grids, ways of forming numbers using given digits to maximise certain products, divisibility tricks, and various other problems.

Perimeter
Perimeter is the total length or total distance covered along the boundary of a closed shape.

Perimeter (P) = AB + BC + CD + DA = a + b + c + d

Area
Area is the total amount of surface enclosed by a closed figure.

Rectangles and Squares

Area of a Rectangle

Area = length (l) × breadth (b)

Area of a Square

Area = Side × Side
= Side²
= a²
where a is the length of each side.

Perimeter Can’t be a Measure of Area
Perimeter cannot be a measure of area for two main reasons:
They are fundamentally different types of measurements with different units, and knowing the perimeter does not uniquely determine the area of a shape.

Triangles
A triangle is a polygon with three sides and three vertices.
It is one of the most basic shapes in geometry.
A triangle with vertices P, Q, and R is denoted by ∆PQR.

Area of a Triangle
The area of a triangle is the region that the triangle occupies in 2D space. The area of different triangles differs based on their size.
If we know the base length and height of a triangle, we can determine its area. It is expressed in square units.
So, the Area of a triangle = \(\frac {1}{2}\) × (Product of base and height of a triangle)

In the triangle PQR, PQ, QR, and RP are the sides.
QR is the triangle’s base, and PS is the triangle’s height.
PS is perpendicular to the side QR from the vertex P.
So, to find the area of ∆PQR, we use the following formula:
Area ∆PQR = \(\frac {1}{2}\) × (Product of base and height of a triangle)
Or, Area ∆PQR = \(\frac {1}{2}\) × (QR × PS)

Areas of different types of triangles
Consider an acute and an obtuse triangle.
Area of each triangle = \(\frac {1}{2}\) × base x height = \(\frac {1}{2}\) × b × h

Visualisation of Area
In the given graph, if the area of each small square is 1 cm², then
Area of rectangle = l × b
= 5 × 2
= 10 cm²

Area of square = a × a
= 2 × 2
= 4 cm²

Area of a Polygon
A polygon is a shape bounded by several straight lines. It can be regular and irregular.
For the regular polygons, it is easy to find the area since the dimensions are definite and known to us.
For example, the area of a rectangle can be easily determined if we know the length of the two adjacent sides.
To determine the area of an irregular polygon, divide the shape into smaller non-overlapping shapes.
The area of the polygon will be equal to the sum of the areas of smaller non-overlapping shapes.

The area of a polygon can be found by dividing it into triangles or using specific formulas depending on the type of polygon.

Parallelogram
A parallelogram is a special kind of quadrilateral.
If a quadrilateral has two pairs of parallel opposite sides, then it is called a parallelogram.

Properties of a parallelogram

• Opposite sides are parallel and congruent.
• Opposite angles are congruent.
• Adjacent angles are supplementary.
• Diagonals bisect each other, and each diagonal divides the parallelogram into two congruent triangles.
• If one of the angles of a parallelogram is a right angle, then all other angles are right, and it becomes a rectangle.

We come across many geometric shapes other than rectangles and squares in our daily lives. Since a few properties of a rectangle and a parallelogram are somewhat similar, the area of the rectangle is similar to the area of a parallelogram.

The area of a parallelogram can be calculated by multiplying its base by its altitude. The base and altitude of a parallelogram are perpendicular to each other, as shown in the following figure.
The formula to calculate the area of a parallelogram can thus be given as,
Area of parallelogram = b × h square units
where b is the length of the base
h is the height or altitude

Let’s do an activity to understand the area of a parallelogram.

Derivation of the Area of a Parallelogram Formula
When we convert the parallelogram into a rectangle

• The base of the parallelogram is equal to the length of the rectangle.
• The height of the parallelogram is equal to the width of the rectangle.

Therefore, the formula for the area of a parallelogram (A) = b × h is derived, where b = base of a parallelogram, h = height of a parallelogram.
After doing this activity, we observed that the area of a rectangle is equal to the area of a parallelogram.
Also, the base and height of the parallelogram are equal to the length and breadth of the rectangle, respectively.

Area of Parallelogram = Base × Height (which is nothing but length × width of the rectangle)

Rhombus
A rhombus is a quadrilateral with all four sides equal.
Its diagonals bisect each other at right angles.

Properties of a Rhombus

• All sides are congruent.
• Opposite angles are congruent.
• The diagonals are perpendicular to and bisect each other.
• Adjacent angles are supplementary (E.g., ∠A + ∠B = 180°).
• A rhombus is a parallelogram whose diagonals are perpendicular to each other.

Area of a Rhombus
Formula: Area = \(\frac {1}{2}\) × d₁ × d₂
Where d₁ is the length of the first diagonal.
d₂ is the length of the second diagonal.

1. Identify the diagonals:
Find the lengths of the two diagonals of the rhombus (the lines connecting opposite corners).
Let them be ‘d₁’ and ‘d₂’

2. Multiply the diagonals:
Multiply the lengths of the two diagonals together (d₁ × d₂)

3. Divide by 2:
Divide the result from Step 2 by 2 to find the area of the rhombus.

Trapezium
A trapezium is a quadrilateral with one pair of parallel sides.

Properties of a Trapezium

• The bases of the trapezium are parallel to each other (MN || OP).
• No sides, angles, or diagonals are congruent.

Area of a Trapezium
Formula: Area = \(\frac {1}{2}\) × (a + b) × h
Where a and b are the lengths of the two parallel sides.
h is the perpendicular distance (height) between the parallel sides.

1. Identify the parallel sides:
Find the two sides of the trapezium that run parallel to each other.
Let their lengths be ‘a’ and ‘b.’

2. Identify the height:
Find the perpendicular distance between these two parallel sides.
This is ‘h’.

3. Sum the parallel sides:
Add the lengths of ‘a’ and ‘b’ together (a + b).

4. Multiply by the height:
Take the sum from Step 3 and multiply it by the height ‘h’: (a + b) × h.

5. Divide by 2:
Divide the result from Step 4 by 2 to get the final area.

Finding the Area using Two Copies of the Trapezium
The area of a trapezium can be found by combining two identical copies to form a parallelogram and then taking half the area of that parallelogram. This geometric derivation leads directly to the standard area formula.

1. Start with one trapezium:
Imagine a trapezium with parallel sides of lengths a and b, and a perpendicular height h.

2. Create a second copy:
Make an identical (congruent) copy of the trapezium.

3. Rotate and combine:
Rotate the second copy by 180 degrees (upside down) and place it next to the first one so that the non-parallel sides align.
The two trapeziums fit together perfectly to form a larger parallelogram.

4. Determine the parallelogram’s dimensions:

• The base of this new parallelogram is the sum of the parallel sides of the original trapezium: a + b
• The height of the parallelogram is the same as the height of the original trapezium: h

5. Calculate the parallelogram’s area:
The area of a parallelogram is given by the formula:
Area = base × height
Area of parallelogram = (a + b) × h

6. Find the trapezium’s area:
Since the parallelogram is made of two identical trapeziums, the area of one trapezium is exactly half the area of the parallelogram.
Area of one trapezium = \(\frac {1}{2}\) × (a + b) × h

The post Algebra Play Class 8 Notes Maths Part 2 Chapter 6 appeared first on Learn CBSE.

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