Category: Mathematics

Introduction

Ratios are said to be equivalent if they can be made simpler or reduced to the same number. In other words, a ratio is said to be equivalent if it can be expressed as a multiple of another ratio.

A ratio can be expressed using a fraction. The concept of an equivalent ratio is comparable to the concept of equivalent fractions. The antecedent and consequent of a ratio can be multiplied or divided by the same number, other than zero, to create an equivalent ratio.

To get a ratio that is equal to the given ratio, we must first represent the ratio in fraction form. Then, by multiplying or dividing the first term and second term by the same non-zero value, the equivalent fraction can be found. We finally convert it to a ratio.

What are Equivalent Ratios

We must first comprehend what equivalence is to understand the equivalence of ratios. Equivalence is very similar to the well-known mathematical relation equal to, as well as to the same mathematical relationship between different objects. In mathematics, equivalence refers to the concept that two objects are equal but distinct because they have the same overall value. When two ratios share the same simplest form, they are said to be equivalent.

Examples of Equivalent Ratios

We can simply create equivalent ratios by multiplying the antecedent and consequent of a ratio by any real number other than the number zero. Thus, creating some examples of equivalent ratios is a very simple task.

For example, we need to find 5 ratios equivalent to 6:10

We can multiply the given ratio by any real number, let’s multiply it by ½

6: 10 = = 3: 5

Thus 3:5 is a ratio equivalent to 6:10

Other such ratios are, 9:15, 12:20, 15:25, 18:30, etc. these all are ratios equivalent to 6:10.

Methods of Finding Equivalent Ratios

There are two methods to find the equivalence of ratios, these methods are

• Cross Multiplication Method

In this method, we multiply the antecedent of the 1st ratio with the consequent of the 2nd ratio and the antecedent of the 2nd ratio with the consequent of the first. If the two products are equal then we can say that the two ratios are equivalent, otherwise, the ratios are not equivalent.

For example: Let’s say we need to use the cross-multiplication method to determine whether the ratios 3:4 and 6:8 are equivalent.

Therefore, we will multiply each ratio’s antecedent by the other ratio’s consequent.

We can say that the ratios are equivalent if the two products are equal.

In this example: 1st Product

3 × 8 = 24

2nd Product

6 × 4 = 24

Since,

Product 1 = Product 2

The ratios 3:4 and 6:8 are equivalent.

• HCF Method

In this method, we first need to represent the ratios as fractions, then we will reduce that fraction into standard form by finding the HCF of the numerator and the denominator, and then dividing the numerator and denominator by that HCF. After both fractions are reduced to the standard form, if they are equal, then the original fraction, i.e., the ratios were equivalent.

For example: Let’s say that we need to use the HCF method to determine whether the ratios 4:14 and 6:21 are equivalent.

To divide the antecedent and consequent of both ratios with their respective HCFs, we will first check the HCF of the antecedent and consequent for each ratio. If the two ratios are equal after the division, the original ratios were equivalent.

In this example: 1st ratio

4:14

HCF (4, 14) = 2

The ratio in the simplest form,

4: 14 == 2: 7

2nd ratio

6: 21

HCF (6, 21) = 3

The ratio in the simplest form

6: 21 = = 2: 7

Since both ratios in their simplest forms are 2:7, thus the original ratios were equivalent.

Use of Equivalent Ratios

In mathematics and other sciences, equivalent ratios have many applications. Some examples of uses are:

• To make the ratios provided simpler.
• We use equivalent ratios to solve any ratio-related problem.
• To calculate ratios between various fractions.
• Various direct proportionality-related scientific issues.

There are also a lot more uses like this.

Read: Applications of Percentage

Summary

We learned about the circumstances under which ratios or proportions are equivalent in this article. We discussed a few instances of equivalent ratios. The following ideas we learned were how to find equivalent ratios. We also discovered how ratios are equivalent using these techniques. Last but not least, we solved several cases that illustrated the concept of equivalent ratios.

Frequently Asked Questions

1. What are Ratios? What are the Components of Ratios?

Ans: Ratios are defined as a comparison between two quantities of the same type. A ratio has 3 parts, 2 parts are the numbers representing the compared quantities antecedent and consequent, and the third part is a symbol, specifically the ‘:’ (colon) symbol, that is put between the two to represent the comparison.

2. Why are Equivalent Ratios Important?

Ans: The equivalent ratios can be used to explain certain relationships between objects in daily life. For instance, if two pens cost Rs. 10, we can use equivalent ratios to determine the price of any other number of pens or the number of pens that can be purchased with a given sum of money. Many other real-world issues can be resolved using equivalent ratios.

3.What are Proportions? What is the Symbol of Proportions?

Ans: Proportions are a comparison between two or more ratios. If two ratios are in proportion, then they are also equivalent.

Introduction

We see many types of shapes in different objects in our daily lives. Geometry is all around us, so much so that we don’t even notice how many named shapes we see or use for our everyday activities. All such geometrical objects have two measuring quantities, which are Area and Perimeter.

The formula for the area and perimeter of a plane figure can be used to calculate the dimensions of two-dimensional shapes. The perimeter is the distance around the edge of the shape, while the area is the space encircled by any closed figure.

This article focuses on the perimeter and its applications in day-to-day life.

Perimeter

The perimeter of any geometric shape is defined as the sum of the length of all the sides of that figure, or in other words, the total length of the boundary around a geometric shape.

The Perimeter of Commonly Known Shapes.

Usually, the perimeter of a shape can be calculated by simply just adding the length of all the sides of the shape. But some geometric shapes have certain properties that make it easier to calculate the perimeter of that shape using a simple formula, such as any Regular polygon, we know regular polygons have all sides equal, so for any n-sided polygon, we can find the perimeter by simply multiplying the side length by n. Some other geometric shapes whose perimeter can be defined using a formula are given in the table below

S. no.	Shape	Perimeter Formula
1.	Equilateral Triangles (with sides a)	3a
2.	Isosceles Triangle (with equal sides a and third side b)	2a+b
3.	Square (with sides a)	4a
4.	Rhombus (with sides a)	4a
5.	Rectangle (with length l, and breadth b)	2(l+b)
6.	Parallelogram (with adjacent sides l and b)	2(l+b)
7.	Kite (with two adjacent sides, a and b)	2(a+b)
8.	Regular n-gon (n-sided polygon) with sides a	n a
9.	Circle (with radius r)	2πr
10.	Semicircle (with radius r)	πr+2r=(π+2)r

The Perimeter of a Combination of Shapes.

Unlike the Area, the perimeter does not just simply add up when two or more shapes are combined, rather the perimeter of the combined shape is usually smaller than the sum of its parts depending on the type of combination.

For Example, take this pentagon-looking shape formed using combining a square and an equilateral triangle of the same side length by their sides.

Here, from the previous section, we know that

Perimeter of Triangle = 3a

Perimeter of Square= 4a

The perimeter of the Combined Shape = 5a < 3a + 4a = 7a

Uses of the Perimeter in Real Life

Perimeter does not quite find any major use in real life, except for measuring boundaries of fields or buildings for fencing purposes. A similar kind of use of fencing is during Christmas when we measure the perimeter of rooms in our house to hang Christmas lights along the walls. Perimeter is used to measure the fences around a protected area, it is also used to calculate the cost of putting the fence. As said before, perimeter does not have a variety in its applications. However, the perimeter can be used to calculate the lateral/curved surface area of 3d objects such as Prisms and Cylinders.

Area

The area is defined as the measure of the region enclosed within the boundaries of a shape.

Solved Examples

Example: What is the perimeter of an isosceles trapezium whose parallel sides are 10 m and 12 m, and the non-parallel sides are 5 m?

Solution: The perimeter of a trapezium has no specific formula; thus, we will simply add all the sides to find the perimeter.

Perimeter = Sum of all the sides

Perimeter = 10 + 12 + 5 + 5 = 32m

Example: What is the perimeter of the following shape, which can be described as a small square cut out from a large square? How does the perimeter of the larger square change from this change in its shape?

Solution: Here, in this image, we can see that the shape is a square on the side of 10 cm, and from it, a smaller square on a side of 5 cm is cut out.

The length of the sides remaining on the larger square is = 10cm – 5cm = 5cm.

Thus, we can calculate the perimeter now, as all the sides are known.

Perimeter = 10cm + 10cm + 5cm + 5cm + 5cm + 5cm = 40cm.

Also, the perimeter of the original large square is 4a = 4 × 10cm = 40cm.

Thus, there is effectively no change in the perimeter of the large square.

Summary

This article discusses the topic of Perimeter, the Perimeter formula for some commonly known shapes, while also shining a light on the uses of the perimeter in daily life. The perimeter of a geometric shape is defined as the length of the boundary of that shape. Perimeter is used to calculate the fencing around objects or places.

Frequently Asked Questions 

1. What do you mean by Perimeter? How will you explain it to someone who has no Mathematical Knowledge?

Ans. Perimeter is defined as the length of the boundary of a geometric shape. To a person who does not know in this regard, we can explain the perimeter as, suppose we take a rope and encircle it tightly along the edges of a shape, then the length of that rope is the perimeter of the shape.

2. What will be the Perimeter of a Square, which has the same area as a Circle with a Radius of r?

Ans. The area is given to be equal, so we will find the length of the side of the square using the areas, and then use that to find the perimeter.

Area of the circle = πr²

Area of a square = a²

Area of the square = Area of the circle

a² = πr²

The perimeter of the square =

3. What is the use of Perimeter in the Calculation of the Lateral Surface Area of any Prism?

Ans. The lateral surface area of a prism is calculated using the formula

LSA = Perimeter of the base × Height

Thus, the perimeter of the base is used to calculate the lateral surface area of a prism.

4. What are the differences between Perimeter and Area?

Perimeter	Area
Perimeter is defined as the measure of the length of the boundary of a geometric shape.	The area is defined as the measure of the 2D region entrapped within the boundaries of a geometric shape.
Perimeter is a 1D quantity.	The area is a 2D quantity.
Perimeter is measured in the unit of length. Thus, its SI unit is m (metres).	The area is measured in the square of the unit of length. Thus, its SI unit is m2 (square metres)

 

Introduction

If all three angles and three sides of one triangle are the same size and dimension as the corresponding angles and sides of the other triangle, then the two triangles are said to be congruent.

The size and shape of any two congruent triangles are the same. Angles in one triangle have a measure that is the same as angles in another triangle. The corresponding sides of the congruent triangles also have equal lengths. Triangles that are congruent with one another can reflect or rotate another.

Congruent Meaning

If two figures can be placed exactly over one another, they are said to be “congruent.”

Take bread example as an example. The bread slices are stacked one on top of the other so that the top slice completely encloses the bottom slice. When stacked on top of the other, every slice of bread is the same size and shape. When something is congruent, it must have the same size and shape. Mathematics uses the term “congruence” to describe when two figures have similar sizes and shapes.

Congruence of Triangles

A triangle is a closed 3-sided figure in geometry. A triangle is a closed polygon made up of three lines that intersect at three different angles. The two triangles are said to be congruent if all three corresponding sides are equal and all three corresponding angles have the same measure. The appearance of these triangles remains constant when they are moved, rotated, flipped, or turned. If the triangles are moved, they must be superimposed to be congruent. As a result, if the triangles’ corresponding sides and angles are equal, the triangles are congruent. Thus, along the corresponding sides and angles, the congruent triangles can be stacked one on top of the other.

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Congruence Criteria for Triangles

Five triangle congruence rules can be used to determine whether a given pair of triangles are congruent or not. The six dimensions allow for a perfect definition of any triangle. The particular triangle has three sides, as well as three angles. As a result, only three of the triangles’ six parts can be used to determine whether two triangles are congruent. The acronym CPCT stands for “Corresponding parts of Congruent triangles”. Triangles can be shown to be congruent, at which point the remaining dimension can be predicted without having to calculate the triangles’ missing sides and angles. The following lists the five triangle congruence rules.

• SSS (Side-Side-Side)
• SAS (Side-Angle-Side)
• ASA (Angle-Side-Angle)
• AAS (Angle-Angle-Side)
• RHS (Right angle -Hypotenuse-Side)

Note: ASA and AAS are not the same. Like ASA and AAS, SAS does not have any related counterpart like SSA, However, a variation of ASS exists for right triangles as RHS, where R is the angle and the other two, i.e., H and S, are sides.

Congruent Triangles Properties – CPCT

Congruent triangles have all their sides and angle equal, thus if two triangles are where we know all the properties of one and not the other, then we can find all the properties of the second triangle by simply comparing the corresponding parts in the first. Congruent triangles are most frequently referred to using the abbreviation CPCT. The term “Corresponding Parts of Congruent Triangles” is abbreviated as CPCT. It is a crucial characteristic of congruent triangles. Congruent triangles’ respective parts are always equal. The congruent triangle properties refer to this.

Types of Congruence

There are 5 types of triangle congruence criteria.

Side Side Side (SSS)

Side Side Side or also known as SSS congruence criteria states that if all three sides of two triangles are equal, then the triangles are congruent.

Side Angle Side (SAS)

Side Angle Side or also known as SAS congruence criteria states that if two sides and their included angles are equal in two triangles, then the triangles are congruent.

Angle Side Angle (ASA)

Angle Side Angle or also known as ASA congruence criteria states that if a side and two angles on it are equal in two triangles, then the triangles are congruent.

Angle Angle Side (AAS)

Angle Angle Side or also known as AAS congruence criteria states that if two angles and a side not common to the two angles are equal in two triangles, then the triangles are congruent.

Right angle Hypotenuse Side (RHS)

Right angle Hypotenuse Side or RHS congruence criteria only applies to right triangles, it says that in two triangles if they have a right angle, and their hypotenuses and one other side are equal in both triangles, then the triangles are congruent.

Summary

In this article, the topic of congruence is discussed in detail. If two figures share the same shape and size, they are said to be congruent; alternatively, if a figure shares the same shape and size as its mirror image, it is said to be congruent to its mirror image.

This article also shines a light on the topic of Rules of congruence for triangles. There are 5 basic congruence criteria, namely SSS, SAS, ASA, AAS, and RHS.

For more help, you can Refer to Lesson 23 congruence of Triangles in Math Class 7.

Frequently Asked Questions

1. What do you mean by Congruence?

Ans 1. Congruent figures are geometric objects that share the same size and shape in mathematics. The two figures are equal and are referred to as congruent figures.

2. Are all Squares Congruent?

Ans 2. No, all squares are not congruent, since for congruence two figures must have all of their quantifying dimensions must be equal, that includes all the sides and all the angles. All squares have the same angles, but their side lengths are different, hence they aren’t congruent.

3. Is AAA a criterion for the Congruence of Triangles?

Ans 3. No, AAA is not a criterion for congruence because even if all the angles of two triangles are equal, that necessarily does not mean that they have the same side length, for example two equilateral triangles of sides 3cm and 5cm, both have the same 60-60-60 angle, but they are not congruent because their sides are of different lengths.