geometry problem solving grade 9

Free Printable regular and irregular polygons Worksheets for 9th Grade

Math teachers, discover a valuable resource for Grade 9 students with our free printable worksheets on regular and irregular polygons. Enhance their understanding and problem-solving skills in geometry.

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Explore printable regular and irregular polygons worksheets for 9th Grade

Regular and irregular polygons worksheets for Grade 9 are essential tools for teachers to help students master the concepts of geometry in Math. These worksheets provide a variety of exercises and problems that challenge students to identify, classify, and analyze different types of polygons. By working through these worksheets, students will gain a deeper understanding of the properties and characteristics of regular and irregular polygons, such as the sum of interior angles, side lengths, and symmetry. Teachers can use these worksheets as a part of their lesson plans, homework assignments, or as supplementary material for students who need extra practice. With a wide range of worksheets available, teachers can easily find the perfect resource to meet the needs of their Grade 9 Math students. Regular and irregular polygons worksheets for Grade 9 are a valuable addition to any geometry curriculum.

Quizizz is an excellent platform for teachers to access a variety of resources, including regular and irregular polygons worksheets for Grade 9 Math students. In addition to worksheets, Quizizz offers interactive quizzes, engaging games, and other educational materials that can be easily integrated into lesson plans. Teachers can create their own quizzes or choose from a vast library of pre-made quizzes, making it simple to assess student understanding of geometry concepts. The platform also provides real-time feedback and analytics, allowing teachers to monitor student progress and identify areas where additional support may be needed. By incorporating Quizizz into their teaching strategies, educators can enhance their Grade 9 Math geometry lessons and provide students with a fun, interactive way to learn about regular and irregular polygons.

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Chapter 13: Geometry of straight lines

In Grade 8 you identified relationships between angles on straight lines. In this chapter, you will revise all of the angle relationships and write clear descriptions of them.

Angle relationships

Remember that 360\(^\circ\) is one full revolution.

If you look at something and then turn all the way around so that you are looking at it again, you have turned through an angle of 360\(^\circ\). If you turn only halfway around, so that you look at something that was right behind your back, you have turned through an angle of 180\(^\circ\).

• Is angle \(\hat{FOD}\) in the figure smaller or bigger than a right angle?
• Is angle \(\hat{FOE}\) in the above figure smaller or bigger than a right angle?

On the figure above, \(\text{F}\)\(\hat{\text{O}}\)\(\text{D}\) + \(\text{F}\)\(\hat{\text{O}}\)\(\text{C}\)= half of a revolution = 180\(^\circ\).

The sum of the angles on a straight line is 180\(^\circ\).

When the sum of angles is 180\(^\circ\), the angles are called supplementary .

• How big is \(\text{C}\)\(\hat{\text{M}}\)\(\text{B}\)?

Why do you say so?

• How big is \(\text{C}\)\(\hat{\text{M}}\)\(\text{P}\)?
• Explain your reasoning.

In the figure below, AMB is a straight line and \(\text{A}\)\(\hat{\text{M}}\)\(\text{C}\) and \(\text{B}\)\(\hat{\text{M}}\)\(\text{C}\) are equal angles.

• How big are these angles?
• How do you know this?

When one line forms two equal angles where it meets another line, the two lines are said to be perpendicular .

Because the two equal angles are angles on a straight line, their sum is 180°, hence each angle is 90°.

In this chapter, you are required to give good reasons for every statement you make.

• Does it look as if \(\text{C}\)\(\hat{\text{M}}\)\(\text{A}\) and \(\text{B}\)\(\hat{\text{M}}\)\(\text{D}\) are equal?
• Can you explain why they are equal?
• Is it true that \(\text{C}\)\(\hat{\text{M}}\)\(\text{A}\) + \(\text{D}\)\(\hat{\text{M}}\)\(\text{A}\) = \(\text{C}\)\(\hat{\text{M}}\)\(\text{A}\) + \(\text{C}\)\(\hat{\text{M}}\)\(\text{B}\)
• Which angle occurs on both sides of the equation in (e)?

Now try to explain your observation in question 5(a).

When two straight lines intersect, the vertically opposite angles are equal.

• If angle BMC = 125\(^\circ\), how big is angle AMD?
• Why do say so?

Lines and angles

A line that intersects other lines is called a transversal .

In the above pattern, AB is parallel to CD and EF \(||\) GH \(||\) KB \(||\) LD.

• Angles a , b , c , d and e are corresponding angles . Do the corresponding angles look appear to be equal?
• Investigate whether the corresponding angles are equal by using tracing paper. Trace the angle you want to compare to other angles and place it on top of the other angle to find out if they are equal. What do you notice?
• Angles f , h , j , m and n are also corresponding angles. Identify all the other groups of corresponding angles in the pattern.
• Describe the position of corresponding angles that are formed when a transversal intersects other lines.

Do these angles appear to be equal?

• Investigate whether the alternate angles are equal by using tracing paper. Trace the angle you want to compare and place it on top of the other angle to find out if they are equal. What do you notice?
• Identify two more pairs of alternate angles.
• Clearly describe the relative position of alternate angles that are formed when a transversal intersects other lines.
• Did you notice something about some of the pairs of corresponding angles when you did the investigation in question 6? Describe your finding.
• Angles f and o ; i and q and k and s are all pairs of co-interior angles . Identify three more pairs of co-interior angles in the pattern.

The angles in the same relative position at each intersection where a straight line crosses two others are called corresponding angles .

Angles on different sides of a transversal and between two other lines are called alternate angles . .

Angles on the same side of the transversal and between two other lines are called co-interior angles . .

Angles formed by parallel lines

Corresponding angles.

The lines AB and CD below never meet. Lines that never meet and are at a fixed distance from one another are called parallel lines. We write AB \(||\) CD.

Parallel lines have the same direction, i.e. they form equal corresponding angles with any line that intersects them.

The line EF cuts AB at G and CD at H.

EF is a transversal that cuts parallel lines AB and CD.

• Look carefully at the angles EGA and EHC in the above figure. They are called corresponding angles . Do they appear to be equal?
• Measure the two angles to check whether they are equal. What do you notice?

When two parallel lines are cut by a transversal, the corresponding angles are equal.

Alternate angles

The angles \(\text{B}\)\(\hat{\text{G}}\)\(\text{F}\) and \(\text{C}\)\(\hat{\text{H}}\)\(\text{E}\) below are called alternate angles . They are on opposite sides of the transversal.

• Do you think angles AGF and DHE should also be called alternate angles?

When parallel lines are cut by a transversal, the alternate angles are equal.

By answering the following questions, you should be able to see how you can explain why alternate angles are equal when parallel lines are cut by a transversal.

What do you know about corresponding angles?

• What can you say about \(\text{B}\)\(\hat{\text{G}}\)\(\text{H}\) + \(\text{A}\)\(\hat{\text{G}}\)\(\text{H}\) Give a reason.
• What can you say about \(\text{D}\)\(\hat{\text{H}}\)\(\text{G}\) + \(\text{C}\)\(\hat{\text{H}}\)\(\text{G}\)? Give a reason.
• Is it true that \(\text{B}\)\(\hat{\text{G}}\)\(\text{H}\) + \(\text{A}\)\(\hat{\text{G}}\)\(\text{H}\) = \(\text{D}\)\(\hat{\text{H}}\)\(\text{G}\) + \(\text{C}\)\(\hat{\text{H}}\)\(\text{G}\)? Explain.
• Will the equation in (c) still be true if you replace angle \(\text{B}\)\(\hat{\text{G}}\)\(\text{H}\) on the left-hand side with angle \(\text{C}\)\(\hat{\text{H}}\)\(\text{G}\)?

Co-interior angles

The angles \(\text{A}\)\(\hat{\text{G}}\)\(\text{H}\) and \(\text{C}\)\(\hat{\text{H}}\)\(\text{G}\) in the figure below are called co-interior angles .

"co-" means together.

"co-interior" means on the same side.

They are on the same side of the transversal.

• What do you know about \(\text{C}\)\(\hat{\text{H}}\)\(\text{G}\) + \(\text{D}\)\(\hat{\text{H}}\)\(\text{G}\)? Explain.
• What do you know about \(\text{B}\)\(\hat{\text{G}}\)\(\text{H}\) + \(\text{A}\)\(\hat{\text{G}}\)\(\text{H}\)? Explain.
• What do you know about \(\text{B}\)\(\hat{\text{G}}\)\(\text{H}\)+ \(\text{C}\)\(\hat{\text{H}}\)\(\text{G}\)? Explain.
• What conclusion can you draw about \(\text{A}\)\(\hat{\text{G}}\)\(\text{H}\) + \(\text{C}\)\(\hat{\text{H}}\)\(\text{G}\)? Give detailed reasons for your conclusion.

When two parallel lines are cut by a transversal, the sum of two co-interior angles is 180\(^\circ\). Another way of saying this is to say that the two co-interior angles are supplementary.

Identify and name angles

• Is RF also perpendicular to CD? Justify your answer.
• Name four pairs of supplementary angles in the figure. In each case say how you know that the angles are supplementary.
• Name four pairs of co-interior angles in the figure.
• Name four pairs of corresponding angles in the figure.
• Name four pairs of alternate angles in the figure.

• If it is also given that RF is perpendicular to AB, will RF also be perpendicular to CD? Justify your answer.
• Name all pairs of supplementary angles in the figure. In each case say how you know that the angles are supplementary.
• Suppose \(\text{E}\)\(\hat{\text{G}}\)\(\text{A}\) = x . Give the size of as many angles in the figure as you can, in terms of x . Each time give a reason for your answer.

When you solve problems in geometry you can use a shorthand way to write your reasons. For example, if two angles are equal because they are corresponding angles, then you can write (corr \(\angle\) s, AB \(||\) CD) as the reason.

• Name five angles in the figure that are equal to \(\text{G}\)\(\hat{\text{H}}\)\(\text{D}\) . Give a reason for each of your answers.
• Name all the angles in the figure that are equal to \(\text{A}\)\(\hat{\text{G}}\)\(\text{H}\) . Give a reason for each of your answers.

Find the sizes of as many angles in the figure as you can, giving reasons.

• Are EF and CD parallel? Give reasons for your answers.

• 9th Grade Math

9th grade math lessons are planned and introduce in different activities. 9th grade math help is provided for the 9th grade students in all segments to cover all the math lesson plans which are categorized into Arithmetic, Algebra, Geometry and Mensuration.

All types of solved examples on different topics are explained along with the step-by-step solutions. 9th grade math practice sheets are arranged in such a way that students can learn math while practicing math problems.

Keeping in mind the mental level of student in ninth grade, every efforts has been made to introduce new concepts in a simple and easy language, so that the students can understand the problems easily.

The difficulty level of the 9th grade math problems has emphasized the theoretical as well as the numerical aspects of the mathematics course. Each topic contains a large number of examples to understand the applications of concepts.

To get prepared for 9th grade math test or exams students need to learn graphing lines on the coordinate plane, solving literal equations, compound inequalities, graphing inequalities in two variables, multiplying binomials, polynomials, factoring techniques for trinomials, solving systems of equations, algebra word problems, variation, rational expressions, rational equations, graphs & functions, circles, construction, triangle theorems & proofs, properties of polygons, transformations, trigonometry and etc……

If student follow math-only-math they can improve their knowledge by practicing the worksheets for 9th graders which will help them to score in their exam.

Ninth Grade Math Lessons – Table of Contents

Rational Numbers

 Decimal Representation of Rational Numbers

Rational Numbers in Terminating and Non-Terminating Decimals

Recurring Decimals as Rational Numbers

Laws of Algebra for Rational Numbers

Comparison between Two Rational Numbers

Rational Numbers Between Two Unequal Rational Numbers

Representation of Rational Numbers on Number Line

Problems on Rational numbers as Decimal Numbers

Problems Based On Recurring Decimals as Rational Numbers

Problems on Comparison Between Rational Numbers

Problems on Representation of Rational Numbers on Number Line

Worksheet on Rational Number as Decimal Numbers

Worksheet on Recurring Decimals as Rational Numbers

Worksheet on Comparison between Rational Numbers

Worksheet on Representation of Rational Numbers on the Number Line

Irrational Numbers

Definition of Irrational Numbers

Decimal Representation of Irrational Number

Representation of Irrational Numbers on The Number Line

Comparison between Two Irrational Numbers

Comparison between Rational and Irrational Numbers

Real number between Two Unequal Real Numbers

Rationalization

Problems on Irrational Numbers

Problems on Rationalizing the Denominator

Worksheet on Irrational Numbers

Profit and Loss

Cost Price, Selling Price and Rates of Profit and Loss

Problems on Cost Price, Selling Price and Rates of Profit and Loss

Understanding Overheads Expenses

Worksheet on Cost Price, Selling Price and Rates of Profit and Loss

Understanding Discount and Mark Up

Successive Discount

Worksheet on Discount and Markup

Worksheet on the application of overhead Expenses

Worksheet on Successive Discounts

Compound Interest

Introduction to Compound Interest

Compound Interest as Repeated Simple Interest

Formulae for Compound Interest

Comparison between Simple Interest and Compound Interest

Worksheet on Compound Interest as Repeated Simple Interest

Worksheet on Use of Formula for Compound Interest

Algebra/Linear Algebra

Expansion of Powers of Binomials and Trinomials

Expansion of (a ± b)^2

Expansion of (a ± b ± c)^2

Expansion of (x ± a)(x ± b)

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

Completing a Square

Simplification of (a + b)(a – b)

Application Problems on Expansion of Powers of Binomials and Trinomials

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

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

Worksheet on Expansion of (x ± a)(x ± b)

Worksheet on Completing Square

Worksheet on Simplification of (a + b)(a – b)

Worksheet on Application Problems on Expansion of Powers of Binomials and Trinomials

Expansion of (a ± b)^3

Simplification of (a ± b)(a^2 ∓ ab + b^2)

Simplification of (a + b + c)(a^2 + b^2 + c^2 – ab – bc – ca)

Expansion of (x + a)(x + b)(x + c)

Problems on Expanding of (a ± b)^3 and its Corollaries

Factorization

Introduction to Factorization

Problems on Factorization by Grouping of Terms

Problems on Factorization of Expressions of the Form a^2 – b^2

Problems on Factorization Using a^2 – b^2 = (a + b)(a – b)

Factorization of a Perfect-square Trinomial

Factorization of Expressions of the Form x^2 + (a + b)x + ab

Factorization of Expressions of the Form ax^2 + bx + c, a ≠ 1

Problems on Factorization of Expressions of the Form x^2 +(a + b)x +ab

Worksheet on Factorization of the Trinomial ax^2 + bx + c

Factorization of Expressions of the Form a^3 + b^3

Factorization of Expressions of the Form a^3 - b^3

Factorization of expressions of the Form a^3 + b^3 + c^3 – 3abc

Factorization of Expressions of the Form a^3 + b^3 + c^3, a + b + c = 0

Miscellaneous Problems on Factorization

Worksheet on Factorization

Linear Equations

Linear Equation in One Variable

Solution of a Linear Equation in One Variable

Laws of Equality

Method of Solving a Linear Equation in One Variable

Problems on Application of Linear Equations

Different Types of Problems in Linear Equation in One Variable

Worksheet on Linear Equation in One Variable

Worksheet on Forming of Linear Equations in One Variable

Worksheet on Solving a Word Problem by using Linear Equation in One Unknown

Changing the Subject of a Formula 

Establishing an Equation

Subject of a Formula

Change of Subject of Formula

Evaluation of Subject by Substitution

Problem on Change the Subject of a Formula

Worksheet on Framing a Formula

Worksheet on Change of Subject

Simultaneous Linear equations

Solution of a Linear Equation in Two Variables

Method of Elimination

Method of Substitution

Method of Cross Multiplication

Exponents/Indices

Power of a Number

Laws of Indices

nth Root of a

Quadratic Equation

Introduction to Quadratic Equation

Formation of Quadratic Equation in One Variable

Solving Quadratic Equations

General Properties of Quadratic Equation

Methods of Solving Quadratic Equations

Roots of a Quadratic Equation

Examine the Roots of a Quadratic Equation

Problems on Quadratic Equations

Quadratic Equations by Factoring

Word Problems Using Quadratic Formula

Examples on Quadratic Equations

Worksheet on Formation of Quadratic Equation in One Variable

Worksheet on Quadratic Formula

Worksheet on Nature of the Roots of a Quadratic Equation

Worksheet on Word Problems on Quadratic Equations by Factoring

Geometry and Measurement

Classification of Triangles on the Basis of Their Sides and Angles

Medians and Altitudes of a Triangle

Geometrical Property of Altitudes

Properties of Angles of a Triangle

Congruency of Triangles

Criteria for Congruency

Problems on Congruency of Triangles

Any point on the bisector of an angle is equidistant from the arms of that angle

An Altitude of an Equilateral Triangle is also a Median

Bisectors of the Angles of a Triangle Meet at a Point

Application of Congruency of Triangles

Angles Opposite to Equal Sides of an Isosceles Triangle are Equal

Equal Sides of an Isosceles Triangle are Produced , the Exterior Angles angles are equal.

The Three Angles of an Equilateral Triangle are Equal.

Sides Opposite to the Equal Angles of a Triangle are Equal

Three Angles of an Equilateral Triangle are Equal

Problems on Properties of Isosceles Triangles

Problem on Two Isosceles Triangles on the Same Base

Lines Joining the Extremities of the Base of an Isosceles Triangle

Points on the Base of an Isosceles Triangle

Theorem on Isosceles Triangle

Inequalities in Triangles

Greater Side has the Greater Angle Opposite to It

Greater Angle has the Greater Side Opposite to It

The Sum of any Two Sides of a Triangle is Greater than the Third Side

Perpendicular is the Shortest Theorem

Comparison of Sides and Angles in a Triangle

Problem on Inequalities in Triangle

Sum Of Any Two Sides Is Greater Than Twice The Median

Sum of the Four Sides of a Quadrilateral Exceeds the Sum of the Diagonals

Midpoint Theorem

Converse of Midpoint Theorem

Four Triangles which are Congruent to One Another

Straight Line Drawn from the Vertex of a Triangle to the Base

Midpoint Theorem on Trapezium

Midsegment Theorem on Trapezium

Midpoint Theorem on Right-angled Triangle

Collinear Points Proved by Midpoint Theorem

Equal Intercepts Theorem

Problems on Equal Intercepts Theorem

Midpoint Theorem by using the Equal Intercepts Theorem 

Proof By the Equal Intercepts Theorem

Enlargement Transformation

Reduction Transformation

Properties of size Transformation

Similar Triangles

Criteria of Similarity between Triangles

AA Criterion of Similarity

Basic Proportionality Theorem

Converse of Basic Proportionality Theorem

Application of Basic Proportionality Theorem

Greater segment of the Hypotenuse is Equal to the Smaller Side of the Triangle

AA Criterion of Similarly on Quadrilateral

Pythagoras’ Theorem

Converse of Pythagoras’ Theorem

Applying Pythagoras’ Theorem

Riders Based on Pythagoras’ Theorem

Rectilinear Figures

Sum of the Interior Angles of an n-sided Polygon

Sum of the Exterior Angles of an n-sided Polygon

Parallelogram

Concept of Parallelogram

Opposite Sides of a Parallelogram are Equal

Opposite Angles of a Parallelogram are Equal

Diagonals of a Parallelogram Bisect each Other

A Quadrilateral is a Parallelogram if its Diagonals Bisect each Other

Pair of Opposite Sides of a Parallelogram are Equal and Parallel

A Rhombus is a Parallelogram whose Diagonals Meet at Right Angles

A Parallelogram whose Diagonals Intersect at Right Angles is a Rhombus

In a Rectangle the Diagonals are of Equal Lengths

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

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

Diagonals of a Parallelogram are Equal & Intersect at Right Angles

Conditions for Classification of Quadrilaterals and Parallelograms

Bisectors of the Angles of a Parallelogram form a Rectangle

Area of a Closed Figure

Base and Height (Altitude) in a Triangle and a Parallelogram

Every Diagonal of a Parallelogram Divides it into Two Triangles of Equal Area

Parallelogram on the Same Base and Between the Same Parallel Lines are Equal in Area

Area of a Parallelogram is Equal to that of a Rectangle Between the Same Parallel Lines

Area of a Triangle is Half that of a Parallelogram on the Same Base and between the Same Parallels

Triangles on the Same Base and between the Same Parallels are Equal in Area

Triangles with Equal Areas on the Same Base have Equal Corresponding Altitudes

Problems on Finding Area of Triangle and Parallelogram

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

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

If Each Diagonal of a Quadrilateral Divides it in Two Triangles of Equal Area then Prove that the Quadrilateral is a Parallelogram

Statistics and Statistical Data

Representation of Data

Statistical Variable

Range of the Statistical Data

Frequency of the Statistical Data

Mean of Ungrouped Data

Arithmetic Mean

Word Problems on Arithmetic Mean

Properties of Arithmetic Mean

Problems Based on Average

Problems on Mean of Ungrouped Data

Properties Questions on Arithmetic Mean

Median of Raw Data

Problems on Median of Ungrouped Data

Worksheet on Mean of Ungrouped Data

Worksheet on Median of Ungrouped Data

Frequency Distribution

Class Interval

Tally Marks

Constructing Frequency Distribution Tables

Class Limits

Class Boundaries

Nonoverlapping Class Intervals into Overlapping Class Intervals

Cumulative Frequency

Mensuration

Plane Figures

Perimeter and Area of Plane Figures

Perimeter and Area of a Rectangle

Geometrical Properties of a Square

Perimeter and Area of a Triangle

Perimeter and Area of Mixed Figures

Perimeter and Area of Rhombus

Perimeter and Area of Parallelogram

Perimeter and Area of Irregular Figures

Perimeter and Area of Regular Hexagon

Area and Circumference of a Circle

Area of a Circular Ring

Area and Perimeter of a Semicircle

Area and Perimeter of Combined Figures

Solid Figures

Cube and Cuboid

Volume and Surface Area of Cuboid

Volume and Surface Area of Cube

Volume and Surface Area of Cube and Cuboid

Volume of Cuboid

Volume of Cube

Lateral Surface Area of a Cuboid

Cross Section

Right Circular Cylinder

Hollow Cylinder

Problems on Right Circular Cylinder

• Probability

Random Experiments

Experimental Probability

Events in Probability

Empirical Probability

Coin Toss Probability

Probability of Tossing Two Coins

Probability of Tossing Three Coins

Complimentary Events

Mutually Exclusive Events

Mutually Non-Exclusive Events

Conditional Probability

Theoretical Probability

Odds and Probability

Playing Cards Probability

Probability and Playing Cards

Probability for Rolling Two Dice

Solved Probability Problems

Probability for Rolling Three Dice

Pre-calculus

Mathematical models with applications, basic trigonometry.

Trigonometry : Beginning of trigonometry and invention of trigonometry and how and why trigonometry is important.

Measurement of Trigonometric Angles : In trigonometry learn about the measurement of angles.

Different systems of measurement of angles and their units.

Sexagesimal System : Concept on understanding Sexagesimal System, problems on Sexagesimal System.

Circular System : Concept on understanding Circular System, problems on Circular System.

Radian is a Constant Angle : Proof that radian is a constant angle.

Relation between Sexagesimal and Circular : Learn about the relation between the units of the two systems.

Conversion from Sexagesimal to Circular System : Learn how to convert or express Sexagesimal to Circular System.

Conversion from Circular to Sexagesimal System : Learn how to convert or express Circular to Sexagesimal Systemm.

Co-ordinate Geometry

Independent Variables and Dependent Variables

Coordinates of a Point

Rectangular Cartesian Coordinates of a Point

Quadrants and Convention for Signs of Coordinates

Plotting a Point in Cartesian Plane

Coordinate Geometry Graph

Graph of Standard Linear Relations Between x, y

Slope of the Graph of y = mx + c

y-intercept of the Graph of y = mx + c

Drawing Graph of y = mx + c Using Slope and y-intercept

Problems on Plotting Points in the x-y Plane

Problems on Slope and Y-intercept

Worksheet on Plotting Points in the Coordinate Plane

Worksheet on Graph of Linear Relations in x, y

Worksheet on Slope and Y-intercept

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9th GRADE MATH PRACTICE PROBLEMS

Problem 1 :

Simplify √20 - √225 + √80

Decompose  20, 225 and 80 into prime factors.

√20  =  √2  ⋅  2  ⋅  5  =  2√5

√225  =  √5  ⋅  5  ⋅  3  ⋅  3  =  5  ⋅  3  =  15

√225  =  √2  ⋅  2  ⋅  2  ⋅  2  ⋅  5  =  (2  ⋅  2)√5  =  4√5

Then, we have

√ 20 -  √ 225 +  √ 80  =   2√5 - 15 + 4√5

√ 20 -  √ 225 +  √ 80  =  6 √5 - 15

√ 20 -  √ 225 +  √ 80   =  6√5 - 15

√ 20 -  √ 225 +  √ 80   =  3(2√5 - 5)

Problem 2 :

The perimeter of the rhombus is 20 cm. One of the diagonals is of length 8 cm. Find the length of the other diagonal.

In a rhombus, the diagonals are intersecting at right angles.

Perimeter of rhombus  =  20 cm

4a  =  20

a  =  5 cm

Length of diagonal  =  8 cm

Half of diagonal  =  4

Let x be the half of the other diagonal.

5 2  =  4 2 + x 2

25  =  16 + x 2

x 2  =  25 - 16

x 2   =  9

x  =  3

Length of other diagonal  =  2(3)

  =  6 cm

Problem 3 :

Jerry and Silvia wanted to go from the southwest corner of a square field to the northeast corner. Jerry walked due east and then due north to reach the goal, but Silvia headed northeast and reached the goal walking in a straight line. How much shorter Silvia's trip was, compared to Jerry's trip? 

Let AB  =  1, BC  =  1

Then AC  =   √1 2 + 1 2

AC  =   √2

Distance covered by Jerry  =  1 + 1  ==> 2

Distance covered by Silvia  =   √2

Difference of distance covered  =  2 -  √2

  =  2 - 1.414

  =  0.586

  =  (0.586/ 2)  ⋅ 100%

  =  30%

So Silvia's trip is 30% shorter than the Jerry's trip.

Problem 4 :

Solve log 10 (2x + 50)  =  3

log 10  (2x + 50) = 3

2x + 50  =  10 3

2x  =  1000-50

2x  =  950

x  =  425

So, the value of x is 425.

Problem 5 :

A set has only one element is called ____________ set 

A set has only one element is called as singleton set.

Problem 6 :

Which of the following statement represents this Venn diagram?

In the given venn diagram, the common region for both A and B is not shaded, then the remaining part of B is shaded. So the correct statement is B-A.

Problem 7: 

If (x+p) (x+q) = x 2 -5x-300, find the value of p² + q²

(x+p) (x+q)   =   x 2 -5x-300

x 2 +(p+q)x+pq  =   x 2 -5x-300

p+q  =  -5 and pq  =  -300

p 2 +q 2   =  (p+q) 2 - 2pq

  =  (-5) 2 -2(-300)

  =  25 + 600

p 2 +q 2   =   625

Problem 8 :

The point of concurrence of the angle bisector of a triangle is called the _____________ of the triangle.

The point of concurrence of the angle bisector of a triangle is called the incenter of the triangle.   9

Problem 9 :

The angles are supplementary and larger angle is twice the smaller angle.Find the angles.

Let x be the smaller angle

2x  -  larger angle

2x+x  =  180

3x   180

x  =  60

So, the required angles are 60 and 120.

Problem 10 :

Find the distance between the points

A (-15,-3) and B (7, 1)

Distance between two points  =  √(7+15) 2 + (1+3) 2

  =  √(-22) 2  + 4 2

  =   √(484  + 16)

  =   √500

  =  10 √5

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How Tesla Planted the Seeds for Its Own Potential Downfall

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