case study of heron's formula class 9th

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CBSE Case Study Questions Class 9 Maths Chapter 12 Heron’s Formula PDF Download

CBSE Case Study Questions Class 9 Maths Chapter 12  are very important to solve for your exam. Class 9 Maths Chapter 12 Case Study Questions have been prepared for the latest exam pattern. You can check your knowledge by solving  case study-based   questions for Class 9 Maths Chapter 12  Heron’s Formula

Case Study Questions Class 9 Maths Chapter 12

Case Study 1: A group of students is learning about Heron’s Formula for finding the area of a triangle. They encountered the following scenario:

Rohan and Kavya came across a triangular field in their village. They made the following observations:

• The lengths of the three sides of the triangular field are 8 meters, 12 meters, and 15 meters.
• The perimeter of the triangular field is 35 meters.

Based on this information, the students were asked to apply Heron’s Formula to find the area of the triangular field. Let’s see if you can answer the questions correctly:

MCQ Questions:

Q1. The semiperimeter of the triangular field is: (a) 8 meters (b) 12 meters (c) 15 meters (d) 17.5 meters

Answer: (d) 17.5 meters

Q2. Using Heron’s Formula, the area of the triangular field is: (a) 24 square meters (b) 30 square meters (c) 36 square meters (d) 40 square meters

Answer: (b) 30 square meters

Q3. The type of triangle formed by the sides of the field is: (a) Equilateral (b) Isosceles (c) Scalene (d) Right-angled

Answer: (c) Scalene

Q4. The length of the altitude corresponding to the side of 15 meters is: (a) 2 meters (b) 4 meters (c) 6 meters (d) 8 meters

Answer: (c) 6 meters

Q5. The lengths of the altitudes corresponding to the sides of 8 meters and 12 meters are: (a) 4 meters and 6 meters (b) 6 meters and 8 meters (c) 8 meters and 10 meters (d) 10 meters and 12 meters

Answer: (a) 4 meters and 6 meters

Case Study 2: A group of students is studying Heron’s Formula for finding the area of a triangle. They encountered the following scenario:

Neha and Mohan went on a field trip to a riverbank. They noticed a triangular piece of land that they wanted to measure and calculate its area. They made the following observations:

• Neha measured the lengths of the three sides of the triangular piece of land as 7 meters, 9 meters, and 11 meters.
• Mohan measured the lengths of the three sides of the same triangular piece of land as 10 meters, 12 meters, and 15 meters.

Based on this information, the students were asked to apply Heron’s Formula to find the area of the triangular piece of land. Let’s see if you can answer the questions correctly:

Q1. Using Neha’s measurements, the semiperimeter of the triangular piece of land is: (a) 13 meters (b) 16 meters (c) 19 meters (d) 23 meters

Answer: (c) 19 meters

Q2. Using Neha’s measurements, the area of the triangular piece of land is: (a) 24 square meters (b) 26 square meters (c) 28 square meters (d) 30 square meters

Answer: (a) 24 square meters

Q3. Using Mohan’s measurements, the semiperimeter of the triangular piece of land is: (a) 16 meters (b) 18 meters (c) 21 meters (d) 25 meters

Answer: (c) 21 meters

Q4. Using Mohan’s measurements, the area of the triangular piece of land is: (a) 40 square meters (b) 42 square meters (c) 45 square meters (d) 48 square meters

Answer: (b) 42 square meters

Q5. The measurements taken by Neha represent a triangle that is: (a) Equilateral (b) Isosceles (c) Scalene (d) Right-angled

Hope the information shed above regarding Case Study and Passage Based Questions for Case Study Questions Class 9 Maths Chapter 12 Heron’s Formula with Answers Pdf free download has been useful to an extent. If you have any other queries about Case Study Questions Class 9 Maths Chapter 12 Heron’s Formula and Passage-Based Questions with Answers, feel free to comment below so that we can revert back to us at the earliest possible.

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Test: Heron`s Formula- Case Based Type Questions - Class 9 MCQ

10 questions mcq test - test: heron`s formula- case based type questions, direction: isosceles triangles were used to construct a bridge in which the base (unequal side) of an isosceles triangle is 4 cm and its perimeter is 20 cm. q. what is the length of equal sides.

So, x + x + 4 = 20

2x + 4 = 20

2x = 20 – 4

x = 16/2 = 8 cm.

Direction: Isosceles triangles were used to construct a bridge in which the base (unequal side) of an isosceles triangle is 4 cm and its perimeter is 20 cm. Q. If the sides of a triangle are in the ratio 3 : 5 : 7 and its perimeter is 300 m. Find its area.

100√2 m 2

500√2 m 2

1500√3 m 2

200√3 m 2

Let the sides of a triangle are a = 3x, b = 5x, c = 7x

Then a + b + c = 300

3x + 5x + 7x = 300

So, a = 60, b = 100, c = 140

= 300/2 = 150

Direction: Isosceles triangles were used to construct a bridge in which the base (unequal side) of an isosceles triangle is 4 cm and its perimeter is 20 cm. Q. What is the Heron's formula for the area of?

{Area} = area

s = semi-perimeter

a = length of side a

b = length of side b

c = length of side c

Isosceles triangles were used to construct a bridge in which the base (unequal side) of an isosceles triangle is 4 cm and its perimeter is 20 cm. What is the semi perimeter of the Isosceles triangle?

Required semi perimeter = Perimeter/2 = 20/2 = 10 m.

Direction: Isosceles triangles were used to construct a bridge in which the base (unequal side) of an isosceles triangle is 4 cm and its perimeter is 20 cm.

Q. What is the area of highlighted triangle ?

Thus, area of the triangle

Direction: Shakshi prepared a Rangoli in triangular shape on Diwali. She makes a small triangle under a big triangle as shown in figure.

Sides of big triangle are 25 cm, 26 cm and 28 cm. Also, ΔPQR is formed by joining mid points of sides of ΔABC.

Use the above data to help her in resolving below doubts.

Q. What is the semi-perimeter of ΔABC?

= (25 + 26 + 28) cm = 79 cm

Q. Area of ΔPQR =

where s is the semi-perimeter of ΔPQR.

Q. ½ of AB =

Q. What is the length of RQ?

Q. If colourful rope is to be placed along the sides of small ΔPQR. What is the length of the rope?

= (12.5 + 13 + 14) cm

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Case Study Questions for Class 9 Maths Chapter 12 Herons Formula

• Last modified on: 2 months ago
• Reading Time: 1 Minute

Here we are providing case study questions for Class 9 Maths Chapter 12 Herons Formula. Students are suggested to solve the questions by themselves first and then check the answers. This will help students to check their grasp on this particular chapter Triangles.

Case Study Questions:

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Chapter 10 Class 9 Herons Formula

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Updated for new NCERT - 2023-24 Curriculmn

Get NCERT Solutions of all exercise questions and examples of Chapter 10 Class 9 Herons Formula. Answers to all question have been solved in a step-by-step manner, with videos of all questions available.

We have studied that

Area of triangle  = 1/2 × Base × Height

In questions where Height and Base is given, we can find the area of triangle easily.

But, in cases where all 3 sides are given, how will we find the area?

If all 3 sides are given, we find Area of Triangle using Herons (or Hero's) Formula

By Hero's Formula

Area of triangle = Square root (s (s-a) (s-b) (s-c))

where a,b, c are sides of the triangle

and s = Semi-Perimeter of Triangle

i.e. s = (a+b+c)/2

In this chapter, we will find Area of Triangle using Herons formula

We will also find Area of Quadrilateral by dividing it into two triangles, and then finding Area of triangle using Hero's Formula

Click on an exercise link or a topic link below to start doing the chapter.

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NCERT Solutions Class 9 Maths Chapter 12 Heron's Formula

NCERT Solutions for Class 9 Maths Chapter 12 Heron’s Formula is a fundamental math concept applied in many fields. Therefore, it is necessary to learn this topic along with understanding its applications. One of the reliable resources to gain this knowledge is by referring to the NCERT Solutions for Class 9 Maths Chapter 12 Heron’s Formula. The solutions are designed in an efficient way to cover these concepts in detail. By practicing questions and sample problems composed in these solutions, students will quickly gain the key skills required for advanced math studies.

There are some ways and formulas to calculate the area of triangles . Heron’s formula is a useful technique to calculate the area of a triangle when the length of all three sides is given. These Class 9 maths NCERT solutions Chapter 12 Heron’s Formula will help students to understand this concept in detail. For more such facts and formulas , read the detailed solution given below and also find some of these in the exercises given below.

• NCERT Solutions Class 9 Maths Chapter 12 Ex 12.1
• NCERT Solutions Class 9 Maths Chapter 12 Ex 12.2

NCERT Solutions for Class 9 Maths Chapter 12 PDF

Triangles are a foundational shape used in various fields of mathematics and other subjects like physics and geography. Learning to find the area of a triangle using Heron’s formula will enable students to gain a better understanding of the related topics and their applications. More details of this topic can be found in the NCERT solutions Class 9 maths chapter 12 Heron's Formula given below:

☛ Download Class 9 Maths NCERT Solutions Chapter 12 Heron’s Formula

NCERT Class 9 Maths Chapter 12   Download PDF

NCERT Solutions for Class 9 Maths Chapter 12 Heron’s Formula

The area of a triangle refers to the space closed within the boundary of a triangle. These solutions will enable students to derive Heron’s formula step-by-step. The concepts explained in these solutions are noteworthy and hold great importance in various spheres of life. The exercise-wise detailed analysis of the NCERT Solutions Class 9 Maths Chapter 12 Heron's Formula is shown below:

• Class 9 Maths Chapter 12 Ex 12.1 - 6 Questions
• Class 9 Maths Chapter 12 Ex 12.2 - 19 Questions

☛ Download Class 9 Maths Chapter 12 NCERT Book

Topics Covered: The topics covered in the Class 9 maths NCERT solutions chapter 12 are as follows: Introduction to Heron’s Formula , Area of a triangle based on its height and base, Area of a triangle using Heron’s Formula, and applications of Heron’s Formula in finding the area of quadrilaterals.

Total Questions: The Class 9 maths chapter 12 Heron's Formula Chapter 12 consists of a total of 15 questions. The students will find some to be easy (3 sums), while others will fall in the moderate (7 sums) and tougher categories (2 sums). 

List of Formulas in NCERT Solutions Class 9 Maths Chapter 12

The NCERT solutions Class 9 maths Chapter 12 involves studying the area of triangles and quadrilaterals , which requires applying different formulas explained in this chapter. Some of the important concepts and formulas listed in this chapter are given below:

• Area of a triangle using Heron’s Formula = A = √{s(s-a)(s-b)(s-c)} , where a, b and c are the length of the three sides of a triangle and s is the semi-perimeter of the triangle given by (a + b + c)/2.
• The area of a quadrilateral whose sides and one diagonal are given can be calculated by

dividing the quadrilateral into two triangles and using Heron’s formula.

Important Questions for Class 9 Maths NCERT Solutions Chapter 12

Video solutions for class 9 maths ncert chapter 12, faqs on ncert solutions class 9 maths chapter 12, why are ncert solutions class 9 maths chapter 12 important.

NCERT Solutions for Class 9 Maths Chapter 12 explains all the concepts and formulas in detail to quickly gain deep knowledge of this chapter. These solutions provide examples, sample problems, and illustrations that are highly beneficial for students to understand the concepts clearly. The solutions are well-organized guides prepared by a team of experts to deliver precise and accurate knowledge of this lesson.

Do I Need to Practice all Questions Provided in NCERT Solutions Class 9 Maths Heron's Formula?

Learning maths requires practice and perseverance. With the practice of a wide range of questions included in the NCERT solutions class 9 maths chapter 12, students will gain knowledge of the core concepts and learn some creative ways to memorize formulas and data. It will enable them to employ their knowledge of particular formulas in various situations, which is highly useful for facing competitive exams.

What are the Important Topics Covered in Class 9 Maths NCERT Solutions Chapter 12?

The important topics covered in the Class 9 maths NCERT solutions chapter 12 are an introduction to triangles, finding the area of a triangle based on its height and base, and calculating the area of a triangle using Heron’s Formula. Additionally, it also covers the topic of applications of Heron’s Formula in finding the area of quadrilaterals.

How Many Questions are there in NCERT Solutions Class 9 Maths Chapter 12 Heron's Formula?

The Class 9 maths chapter 12 Heron's Formula Chapter 12 consists of a total of 15 questions. All of them are based on this formula and its applications. Therefore, the students should practice and memorize it carefully.

How CBSE Students can utilize NCERT Solutions Class 9 Maths Chapter 12 effectively?

NCERT Solutions Class 9 Maths Chapter 12 comprises interactive illustrations and exercises that will assist students in gaining a better understanding of Heron's Formula in practical situations. The students should read the entire chapter carefully as each concept included in these solutions is vital for understanding facts and formulas applied in this lesson.

Why Should I Practice Class 9 Maths NCERT Solutions Heron's Formula chapter 12?

The NCERT Solutions Class 9 Maths Heron's Formula chapter 12 has been prepared by experts in their respective fields to deliver comprehensive knowledge of each and every concept. The facts and data incorporated in these solutions are compiled to promote accurate knowledge in an easy-to-understand manner. Thus, it is very necessary to practice all questions in the NCERT textbook.

NCERT Solutions for Class 9 Maths Chapter 12 Heron's Formula

Ncert solutions for class 9 maths chapter 12 heron's formula| pdf download.

• Exercise 12.1 Chapter 12 Class 9 Maths NCERT Solutions
• Exercise 12.2 Chapter 12 Class 9 Maths NCERT Solutions

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Unit 10: Heron's Formula

Area of triangle - by heron's formula.

• Heron's formula (Opens a modal)
• Heron's Formula 10.1 4 questions Practice
• Finding area of triangle using Heron's formula 4 questions Practice

NCERT Solutions for Class 9 Maths Chapter 12 Heron’s Formula

NCERT Solutions for Class 9 Maths Chapter 12 Heron’s Formula are provided here. Our NCERT Maths solutions contain all the questions of the NCERT textbook that are solved and explained beautifully. Here you will get complete NCERT Solutions for Class 9 Maths Chapter 12 all exercises Exercise in one place. These solutions are prepared by the subject experts and as per the latest NCERT syllabus and guidelines. CBSE Class 9 Students who wish to score good marks in the maths exam must practice these questions regularly.

Class 9 Maths Chapter 12 Heron’s Formula NCERT Solutions

Below we have provided the solutions of each exercise of the chapter. Go through the links to access the solutions of exercises you want. You should also check out our NCERT Class 9 Solutions for other subjects to score good marks in the exams.

NCERT Solutions for Class 9 Maths Chapter 12 Exercise 12.1

NCERT Solutions for Class 9 Maths Chapter 12 Exercise 12.2

NCERT Solutions for Class 9 Maths Chapter 12 – Topic Discussion

Below we have listed the topics that have been discussed in this chapter.

• Heron’s Formula

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Heron's Formula Questions

Heron’s formula questions with answers are provided here. Class 9 students can practise the questions based on Heron’s formula to prepare for the exams. These extra questions are prepared by our subject experts, as per the NCERT curriculum and latest CBSE syllabus (2022-2023). Learn Heron’s formula in detail at BYJU’S.

Definition: Heron’s formula is a formula used to find the area of a triangle. According to this formula;

Area of triangle = √(s(s-a)(s-b)(s-c))

Where a, b and c are the sides of a triangle and s is the semiperimeter of triangle.

s = (a+b+c)/2

Heron’s Formula Questions and Solutions

Q.1: Find the area of a triangle whose sides are 12 cm, 6 cm and 15 cm.

Solution: Given the sides of a triangle are:

According to Heron’s formula;

s = (12 + 6 + 15)/2 = 33/2 = 16.5

Area = √(16.5(16.5-12)(16.5-6)(16.5-15))

= √(16.5 x 4.5 x 0.5 x 1.5)

= 34.2 cm 2

Q.2: Find the Area of a Triangle whose two sides are 18 cm and 10 cm, respectively and the perimeter is 42 cm.

Solution: Given two sides of a triangle are 18 cm and 10 cm, respectively.

a = 18cm, b = 10 cm

Perimeter of triangle = 42 cm

18+10+c = 42

c = 42 – 28 = 14 cm

Using Heron’s formula, we have;

Semiperimeter, s = 42/2 = 21

Area = √(21(21-18)(21-10)(21-14))

= √(21 x 3 x 11 x 7)

= 69.7 cm 2

Q.3: A triangular park has sides 120 m, 80 m and 50 m. A gardener has to put a fence all around it and also plant grass inside. How much area does he need to plant?

Solution: Given,

Sides of triangular park are 120m, 80m and 50m.

Semiperimeter, s = (120 + 80 + 50)/2 = 125 m

Area = √(125 (125-120) (125 – 80) (125 – 50)

= √(125 x 75 x 45 x 5)

= 375√15 m 2

Q.4: The sides of a triangle are in the ratio of 12: 17: 25 and its perimeter is 540 cm. Find its area.

Ratio of the sides of the triangle is 12: 17: 25

Let the sides of triangle be 12x, 17x and 25x

Given, perimeter of the triangle = 540 cm

12x + 17x + 25x = 540 cm

⇒ 54x = 540cm

Thus, the sides of the triangle are:

12 x 10 = 120 cm

17 x 10 = 170 cm

25 x 10 = 250 cm

Semiperimeter, s = 540/2 = 270 cm

Using Heron’s formula,

= √(270 x 150 x 100 x 20)

= 9000 cm 2

Q.5: Find the area of a triangle whose sides are 4.5 cm and 10 cm and perimeter 20.5 cm.

Side a = 4.5 cm

Side b = 10 cm

Perimeter of triangle = 20.5 cm

a+b+c = 20.5

4.5+10+c = 20.5

14.5+c = 20.5

c = 20.5 – 14.5

Semiperimeter, s = (4.5+10+6)/2 = 20.5/2 = 10.25 cm

Area = 7.91 cm 2

Q.6: What is the area of a triangle whose sides are 9 cm, 12 cm and 15 cm?

Solution: Given, the sides of a triangle are:

Semiperimeter, s = (9 + 12 + 15)/2 = 36/2 = 18 cm

Area = 54 cm 2

Q.7: The perimeter of a right triangle is 300m. If its sides are in the ratio 3 : 5 : 7. Find the area of the triangle.

Perimeter of right triangle = 300 m

Ratio of sides of triangle is 3 : 5 : 7

Let the sides of triangle be:

3x + 5x + 7x = 300

x = 300/15 = 20

Thus, the sides of triangle are:

a = 3x = 3 (20) = 60 m

b = 5x = 5 (20) = 100 m

c = 7x = 7 (20) = 140 m

Semiperimeter, s = 300/2 = 150 m

Area = √(150 (150 – 60) (150 – 100) (150 – 140))

Area = 1500√3 m 2

Q.8: The sides of a triangle are 7 cm, 9 cm, and 14 cm. What is the area of the triangle?

a = 7cm, b = 9 cm and c = 14 cm

Semiperimeter, s = (7 + 9 + 14)/2 = 15 cm

Area of triangle = = √[s (s-a) (s-b) (s-c)] By Heron’s formula.

= 12√5 cm 2

Q.9: The sides of a triangle are 11 m, 60 m and 61 m. What is the altitude to the smallest side?

Solution: Given, sides of the triangle are 11 m, 60 m and 61 m.

The smallest side is 11 m.

Area of triangle = ½ (base) (height)

A = ½ (11) h ….(i)

We can find the area of the triangle using Heron’s formula here.

s = (11+60+61)/2 = 66 m

Area = √(66 x 55 x 6 x 5) = 330 m 2

Now putting the value of area in equation (i), we get;

330 = ½ (11) h

h = (2 x 330)/11 = 60m

Therefore, the altitude to the smallest side is 60m.

Q.10: If every side of a triangle is doubled, by what percentage is the area of the triangle increased?

Solution: Let a, b and c be the sides of a triangle.

Semiperimeter, s = (a+b+c)/2

Now, if each of the side is doubled, then the new sides of a triangle are:

A = 2a, B = 2b, C = 2c

Semiperimeter, S = (A+B+C)/2 = (2a + 2b + 2c)/2 = 2s

By Heron’s formula,

= 4√s(s-a)(s-b)(s-c)

Increase in Area = (4A – A)/A x 100% = 300%

Hence, the area is increased by 300%, if the sides of the triangle are doubled.

Related Articles

• Class 9 Maths MCQs Chapter 12 Heron’s Formula
• Heron’s Formula Class 9 Notes: Chapter 12
• Important Questions Class 9 Maths Chapter 12-Heron’s Formula

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