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Saturday, September 9, 2023

QUESTION BANK CLASS 8 UNDERSTANDING QUADRILATERALS

A simple closed curve made up of only line segments is called a polygon.
Polygons that are convex have no portions of their diagonals in their exteriors or any line segment joining any two different points, in the interior of the polygon, lies wholly in the interior of it
A regular polygon is both ‘equiangular’ and ‘equilateral’. For example, a square has sides of equal length and angles of equal measure. Hence it is a regular polygon.
A rectangle is equiangular but not equilateral.
the sum of the measures of the external angles of any polygon is 360°
Trapezium is a quadrilateral with a pair of parallel sides
Kite is a special type of a quadrilateral. The sides with the same markings in each figure are equal. For example AB = AD and BC = CD.
A kite has 4 sides (It is a quadrilateral). (ii) There are exactly two distinct consecutive pairs of sides of equal length.
A parallelogram is a quadrilateral. As the name suggests, it has something to do with parallel lines.
A parallelogram is a quadrilateral whose opposite sides are parallel.
There are four sides and four angles in a parallelogram.
The opposite sides of a parallelogram are of equal length.
The opposite angles of a parallelogram are of equal measure.
The diagonals of a parallelogram, in general, are not of equal length.
The diagonals of a parallelogram bisect each other.
A rhombus is a quadrilateral with sides of equal length. Since the opposite sides of a rhombus have the same length, it is also a parallelogram. So, a rhombus has all the properties of a parallelogram and also that of a kite.
The diagonals of a rhombus are perpendicular bisectors of one another.
A rectangle is a parallelogram with equal angles.
a rectangle is a parallelogram in which every angle is a right angle. Being a parallelogram, the rectangle has opposite sides of equal length and its diagonals bisect each other.
The diagonals of a rectangle are of equal length.
In a rectangle the diagonals, besides being equal in length bisect each other.
A square is a rectangle with equal sides.
In a square the diagonals.
(i) bisect one another (square being a parallelogram)
(ii) are of equal length (square being a rectangle) and
(iii) are perpendicular to one another.
The diagonals of a square are perpendicular bisectors of each other

POINTS TO REMEMBER

Parallelogram:

A quadrilateral with each pair of opposite sides parallel.
(1) Opposite sides are equal. (2) Opposite angles are equal. (3) Diagonals bisect one another.

Rhombus:

A parallelogram with sides of equal length.
(1) All the properties of a parallelogram. (2) Diagonals are perpendicular to each other.

Rectangle: A parallelogram with a right angle

(1) All the properties of a parallelogram. (2) Each of the angles is a right angle. (3) Diagonals are equal.

Square: A rectangle with sides of equal length.

All the properties of a parallelogram, rhombus and a rectangle.

Kite: A quadrilateral with exactly two pairs of equal consecutive sides

(1) The diagonals are perpendicular to one another (2) One of the diagonals bisects the other. (3) In the figure m∠B = m∠D but m∠A ≠ m∠C.

EXERCISE 3.1

1. Given here are some figures

Classify each of them on the basis of the following. (a) Simple curve (b) Simple closed curve (c) Polygon (d) Convex polygon (e) Concave polygon

2. What is a regular polygon? State the name of a regular polygon of (i) 3 sides (ii) 4 sides (iii) 6 sides

EXERCISE 3.2

1. Find x in the following figures.

2. Find the measure of each exterior angle of a regular polygon of (i) 9 sides (ii) 15 sides
3. How many sides does a regular polygon have if the measure of an exterior angle is 24°?
4. How many sides does a regular polygon have if each of its interior angles is 165°?
5. (a) Is it possible to have a regular polygon with measure of each exterior angle as 22°?
(b) Can it be an interior angle of a regular polygon? Why?
6. (a) What is the minimum interior angle possible for a regular polygon? Why?
(b) What is the maximum exterior angle possible for a regular polygon?

EXERCISE 3.3

1. Given a parallelogram ABCD. Complete each statement along with the definition or property used. (i) AD = ...... (ii) ∠ DCB = ...... (iii) OC = ...... (iv) m ∠DAB + m ∠CDA = ......

2. Consider the following parallelograms. Find the values of the unknowns x, y, z.

3. Can a quadrilateral ABCD be a parallelogram if (i) ∠D + ∠B = 180°? (ii) AB = DC = 8 cm, AD = 4 cm and BC = 4.4 cm? (iii) ∠A = 70° and ∠C = 65°?

4. Draw a rough figure of a quadrilateral that is not a parallelogram but has exactly two opposite angles of equal measure.

5. The measures of two adjacent angles of a parallelogram are in the ratio 3 : 2. Find the measure of each of the angles of the parallelogram.

6. Two adjacent angles of a parallelogram have equal measure. Find the measure of each of the angles of the parallelogram.

7. The adjacent figure HOPE is a parallelogram. Find the angle measures x, y and z. State the properties you use to find them.

8. The following figures GUNS and RUNS are parallelograms. Find x and y. (Lengths are in cm)

9. In the above figure both RISK and CLUE are parallelograms. Find the value of x.

10. Explain how this figure is a trapezium. Which of its two sides are parallel? (Fig)

11. Find m∠C in Fig, if

\bar{A B} ∥ \bar{D C}

12. Find the measure of ∠P and ∠S if

\bar{S P} ∥ \bar{R Q}

in Fig. (If you find m∠R, is there more than one method to find m∠P?)

EXERCISE 3.4

1. State whether True or False.
(a) All rectangles are squares

(e) All kites are rhombuses.

(b) All rhombuses are parallelograms

(f) All rhombuses are kites.

(g) All parallelograms are trapeziums.

(d) All squares are not parallelograms.

(h) All squares are trapeziums.

2. Identify all the quadrilaterals that have.

(a) four sides of equal length

(b) four right angles

3. Explain how a square is.

(i) a quadrilateral

(ii) a parallelogram

(iii) a rhombus

(iv) a rectangle

4. Name the quadrilaterals whose diagonals.

(i) bisect each other

(ii) are perpendicular bisectors of each other

(iii) are equal

5. Explain why a rectangle is a convex quadrilateral.

6. ABC is a right-angled triangle and O is the mid point of the side opposite to the right angle. Explain why O is equidistant from A, B and C. (The dotted lines are drawn additionally to help you).

EXAMPLE

Find the number of sides of a regular polygon whose each exterior angle has a measure of 45°.
RENT is a rectangle (Fig 3.35). Its diagonals meet at O. Find x, if OR = 2x + 4 and OT = 3x + 1.
RICE is a rhombus (Fig 3.30). Find x, y, z. Justify your findings.
In Fig, HELP is a parallelogram. (Lengths are in cms). Given that OE = 4 and HL is 5 more than PE? Find OH.

In a parallelogram RING, (Fig 3.22) if m∠R = 70°, find all the other angles

In Fig, BEST is a parallelogram. Find the values x, y and z.

Find the perimeter of the parallelogram PQRS

Find measure x in Fig.

QUESTION BANK CLASS 8 LINEAR EQUATIONS WITH ONE VARIABLES

An algebraic equation is an equality involving variables. It has an equality sign. The expression on the left of the equality sign is the Left Hand Side (LHS). The expression on the right of the equality sign is the Right Hand Side (RHS).
In an equation the values of the expressions on the LHS and RHS are equal. This happens to be true only for certain values of the variable. These values are the solutions of the equation.

EXAMPLE

Solve 2x – 3 = x + 2

EXERCISE 2.1

Solve the following equations and check your results

1. 3x = 2x + 18

2. 5t – 3 = 3t – 5

3. 5x + 9 = 5 + 3x

4. 4z + 3 = 6 + 2z

5. 2x – 1 = 14 – x

6. 8x + 4 = 3 (x – 1) + 7

EXERCISE 2.2

Solve the following linear equations.

Simplify and solve the following linear equations.

7. 3(t – 3) = 5(2t + 1)

8. 15(y – 4) –2(y – 9) + 5(y + 6) = 0

9. 3(5z – 7) – 2(9z – 11) = 4(8z – 13) – 17

10. 0.25(4f – 3) = 0.05(10f – 9)

POINTS TO REMEMBER

1. An algebraic equation is an equality involving variables. It says that the value of the expression on one side of the equality sign is equal to the value of the expression on the other side.

2. The equations we study in Classes VI, VII and VIII are linear equations in one variable. In such equations, the expressions which form the equation contain only one variable. Further, the equations are linear, i.e., the highest power of the variable appearing in the equation is 1.

3. An equation may have linear expressions on both sides. Equations that we studied in Classes VI and VII had just a number on one side of the equation.

4. Just as numbers, variables can, also, be transposed from one side of the equation to the other.

5. Occasionally, the expressions forming equations have to be simplified before we can solve them by usual methods. Some equations may not even be linear to begin with, but they can be brought to a linear form by multiplying both sides of the equation by a suitable expression.

6. The utility of linear equations is in their diverse applications; different problems on numbers, ages, perimeters, combination of currency notes, and so on can be solved using linear equations.

QUESTION BANK CLASS 8 RATIONAL NUMBERS

EXERCISE 1.1

1. Name the property under multiplication used in each of the following

2. Tell what property allows you to compute

3. The product of two rational numbers is always a ___.

POINTS TO REMEMBER

A number which can be written in the form p q , where p and q are integers and q ≠ 0 is called a rational number
rational numbers are closed under addition. That is, for any two rational numbers a and b, a + b is also a rational number.
rational numbers are closed under subtraction. That is, for any two rational numbers a and b, a – b is also a rational number.
rational numbers are closed under multiplication. That is, for any two rational numbers a and b, a × b is also a rational number.
for any rational number a, a ÷ 0 is not defined. So rational numbers are not closed under division.
two rational numbers can be added in any order. We say that addition is commutative for rational numbers. That is, for any two rational numbers a and b, a + b = b + a.
subtraction is not commutative for integers and integers are also rational numbers. So, subtraction will not be commutative for rational numbers too.
multiplication is commutative for rational numbers. In general, a × b = b × a for any two rational numbers a and b.
division is not commutative for rational numbers.
addition is associative for rational numbers. That is, for any three rational numbers a, b and c, a + (b + c) = (a + b) + c.
Subtraction is not associative for rational numbers.
multiplication is associative for rational numbers. That is for any three rational numbers a, b and c, a × (b × c) = (a × b) × c.
division is not associative for rational numbers.
Zero is called the identity for the addition of rational numbers. It is the additive identity for integers and whole numbers as well.
a × 1 = 1 × a = a for any rational number a
1 is the multiplicative identity for rational numbers.
Distributivity of Multiplication over Addition and Subtraction. For all rational numbers a, b and c, a (b + c) = ab + ac a (b – c) = ab – ac

1. Rational numbers are closed under the operations of addition, subtraction and multiplication.

2. The operations addition and multiplication are (i) commutative for rational numbers. (ii) associative for rational numbers.

3. The rational number 0 is the additive identity for rational numbers.

4. The rational number 1 is the multiplicative identity for rational numbers.

5. Distributivity of rational numbers:

For all rational numbers a, b and c, a(b + c) = ab + ac and a(b – c) = ab – ac

6. Between any two given rational numbers there are countless rational numbers.

The idea of mean helps us to find rational numbers between two rational numbers.