It is often said that “Mathematics is the the mother of science”. Infact, Math’s is a great large dig of knowledge that contains chunks of almost everything happening near around us in every field of life. Either if it is a physics or a medical field, mathematics plays its role every time.
In this article, we will learn about some main topics of math’s problem solving techniques like parallel lines & perfect squares. So lets have a look below:
Two or more lines that fall in the same non-intersecting plane are called parallel lines. They are equal to each other and have the same slope. Let’s learn more about parallel lines in this article.
What is a parallelogram? Parallel lines are straight lines that never meet, no matter how long we stretch them. Look at the following figure which shows similar lines. Line ‘a’ corresponds to line ‘b’ and line ‘p’ corresponds to line ‘q’.
The Parallel Lines and Transverse
When two lines are joined by another line called a secant, multiple pairs of lines are created. While some parts are compatible (equal), others are complementary. Look at these figures to see the lines labeled L1 and L2 that are cut transversally. Eight different angles are formed by two parallel lines and one rotation. Each section has been labeled using a letter.
Properties of Parallel Lines
Similar lines can be easily identified by the main characteristics shown below.
- Parallel lines are straight lines that are the same distance from each other.
- Parallel lines do not meet, no matter how far they are extended in any direction.
Similar properties are followed by online parallel line calculator to get the solution of problems related to this concept of maths.
How do you know if the Lines are Parallel?
Besides the characteristics given above, when any two lines are connected transversally, they can be distinguished by the following characteristics.
- Any two straight lines are said to be congruent if their corresponding sides are equal.
- Any two straight lines are said to be congruent if their corresponding interior angles are the same.
- Any two straight lines are said to be congruent if the other angles drawn from it are equal.
- Any two straight lines are said to be congruent if consecutive interior angles of the same secant are additive.
Parallel Lines Equation
The equation of a straight line is usually written in the intercept-slope form represented by the equation, y = mx + b, where ‘m’ is the slope and ‘b’ is the ordinate at the origin. The ‘m’ value determines the slope or gradient and tells us the slope of the line.
It should be noted that the slope of two parallel lines is usually the same. For example, if the slope of a line equal to y = 4x + 3 is 4. Therefore, any line equal to y = 4x + 3 will have the same slope, which is 4. The parallel lines have different overlaps and have no common points.
Symbol of the Parallel Lines
Parallel lines are lines that never meet, no matter how long we stretch them. The symbol used to indicate parallel lines is ||. For example, AB II PQ shows that line AB is in line PQ. The symbol for odd rows is ∦.
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A perfect square is a number that can be expressed as the product of an integer by itself or as the second factor of an integer. For example, 25 is a perfect square because it is the product of the integer 5 and itself, 5 × 5 = 25. However, 21 is not a perfect square number because it cannot be defined as the product of two similar integers.
In this section, we will discuss the concept of perfect squares and learn how to recognize them. We will discuss the definition of perfect square, its formula and list of a perfect squares and some solved examples for better understanding.
What is a perfect square?
A perfect square is a positive number obtained by multiplying an integer by itself. In simple terms, we can say that a perfect square is a number that is the product of all the numbers themselves. In general, we can express a perfect square as x2, where x is an integer and the value of x2 is a perfect square.
However, if you are confused that the number is a perfect square or not, then must try the perfect square calculator, this will tell you whether the number is a perfect square or not?
Definition of a perfect square
A perfect square is a number obtained by multiplying an integer or a whole number. Let’s take an example to understand the concept behind the perfect square. For this we can take another set of 4 balls and 6 balls. Arrange the marbles. Can you arrange the marbles as shown below?
Let’s review this work. We can form a square with 4 such marbles that there are 2 rows, and 2 marbles in each row. With 6 marbles, we can form a rectangle that has 2 rows, and 3 marbles in each row. Mathematically, this means 4 = 2 × 2 and 6 = 3 × 2. Let’s focus on the numbers that form a square. Here, 4 = 2 × 2 = 22. Now, if we look at the definition of a perfect square, it says, “A perfect square is the number obtained by squaring an integer.
Perfect square formula
Suppose that N is the perfect square of an integer x, which can be written as N = product of x and x = x2. Therefore, the perfect square formula can be defined as follows:
Let’s replace the method with a pattern. If x = 9, then N = x2. This means that N=92=81. Here, 81 is a perfect square because it is the square of the integer, 9. This can be understood in another way using the square root. To know whether a number is a perfect square or not, we calculate the square root of a given number.
If the square root is an integer then the given number is a perfect square, and if the value of the square root is not an integer then the given number is not a perfect square.
For example, to check whether 21 is a perfect square or not, let’s calculate its square root. √21 = 4.58. As we can see, 4.58 is not a whole number/number, so 21 is not a perfect square number.
Let us take another example of the number 64 → √64 = 8. We can see that 8 is a number, so 64 is a perfect square. Now that we know perfect squares, let’s learn perfect square trinomials.
Perfect Square Trinomial:
The term obtained from the square of a binomial is called a perfect square trinomial. For example, if we divide the term (y+3), we use the property, (a+b)2= a2 +2ab+b2, and get, (y+3)2 = y2 +6y +9. Here, y2 +6y+9 is a perfect square trinomial. Other examples of perfect square trinomials are y2 -8y+16 and 4×2+ 12x +9.
How do you identify a perfect square?
Place the last number of perfect square numbers from 1 to 20 as shown in the table above. You will notice that they end in one of these numbers 0, 1, 4, 5, 6 or 9. When you have tested enough different numbers, you will have found something important. of a perfect square. Numbers that have any of the digits 2, 3, 7, or 8 and their denominators are not perfect square numbers, while numbers that have any digits of 0, 1, 4, 5, 6, or 9 where they can. are perfect squares. The following observations can be made to determine the perfect square.
- Numbers ending in 3 and 7 will have 9 because the units keep the numbers in their square numbers.
- A number ending in 5 will have 5 because its unit is the same number as its square number.
- Numbers ending in 4 and 6 will have 6 because those units keep the number in its square number.
- Numbers ending in 2 and 8 will have 4 because those units keep the number in its square number.
- Numbers ending in 1 and 9 will have 1 because those units keep the number in their square number.
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Let’s look at some of the differences from the above rule of perfect square numbers. The numbers 159 and 169 end with the number 9, but 169 is a perfect square, while 159 is not. If the number ends with the number 0, then you can look for the following: Is there a zero at the end of the number? Let’s say we have 1000 numbers. If there is a non-zero number, then it is not a perfect square. 1000 has 3 zeros at the end. So it’s not a perfect square. If there is a zero number, it can be a perfect square. 400 and 300 both have a number of zeros, but 400 = 202, which is a perfect square, but 300 is not a square of an integer.
Another way to find perfect squares
Another way to check whether a number is a perfect square or not is to calculate the square root of a given number. If the square root is a number, then it is a perfect square. If the square root is not a number, then the given number is not a perfect square. For example, to check whether 24 is a perfect square or not, let’s calculate its square root. √24 = 4.89. As we can see, 4.89 is not a perfect number, so 24 is not a perfect square. Let’s take another example of the number 81. √81 = 9. We can see that 9 is a number, so 81 is a perfect square.