(e) The Sides Of A Triangle Are 9 Cm, 12 Cm And 15 Cm; The Area Of The Triangle Is : (i) 54 Cm² (iii) 108 Cm² (ii) 96 Cm² (iv) 135 Cm²
Introduction
In this article, we will delve into the world of geometry and explore the concept of finding the area of a triangle. Given the sides of a triangle, we will use Heron's formula to calculate the area. We will also discuss the different possible scenarios and their corresponding solutions.
Understanding Heron's Formula
Heron's formula is a mathematical formula used to find the area of a triangle when all three sides are known. The formula is as follows:
A = √(s(s-a)(s-b)(s-c))
where A is the area of the triangle, and s is the semi-perimeter, which is calculated as:
s = (a + b + c) / 2
where a, b, and c are the lengths of the sides of the triangle.
Applying Heron's Formula to the Given Triangle
Let's apply Heron's formula to the given triangle with sides 9 cm, 12 cm, and 15 cm.
Step 1: Calculate the Semi-Perimeter
First, we need to calculate the semi-perimeter of the triangle.
s = (9 + 12 + 15) / 2 s = 36 / 2 s = 18
Step 2: Apply Heron's Formula
Now that we have the semi-perimeter, we can apply Heron's formula to find the area of the triangle.
A = √(18(18-9)(18-12)(18-15)) A = √(18(9)(6)(3)) A = √(2916) A = 54
Analyzing the Results
We have found that the area of the triangle is 54 cm². However, we are given four possible answers: 54 cm², 96 cm², 108 cm², and 135 cm². Let's analyze each of these options to determine which one is correct.
Option (i): 54 cm²
As we have already calculated, the area of the triangle is indeed 54 cm². This option is correct.
Option (ii): 96 cm²
To determine if this option is correct, we need to recalculate the area using Heron's formula.
A = √(s(s-a)(s-b)(s-c)) A = √(18(18-9)(18-12)(18-15)) A = √(18(9)(6)(3)) A = √(2916) A = 54
This option is incorrect, as the area of the triangle is not 96 cm².
Option (iii): 108 cm²
To determine if this option is correct, we need to recalculate the area using Heron's formula.
A = √(s(s-a)(s-b)(s-c)) A = √(18(18-9)(18-12)(18-15)) A = √(18(9)(6)(3)) A = √(2916) A = 54
This option is incorrect, as the area of the triangle is not 108 cm².
Option (iv): 135 cm²
To determine if this option is correct, we need to recalculate the area using Heron's formula.
A = √(s(s-a)(s-b)(s-c)) A = √(18(18-9)(18-12)(18-15)) A = √(18(9)(6)(3)) A = √(2916) A = 54
This option is incorrect, as the area of the triangle is not 135 cm².
Conclusion
In conclusion, the correct answer is option (i): 54 cm². We have used Heron's formula to calculate the area of the triangle and have found that it is indeed 54 cm².
Additional Tips and Tricks
- When using Heron's formula, make sure to calculate the semi-perimeter correctly.
- Double-check your calculations to ensure that you are getting the correct answer.
- If you are given multiple options, recalculate the area using Heron's formula to determine which option is correct.
Final Thoughts
Q: What is Heron's formula?
A: Heron's formula is a mathematical formula used to find the area of a triangle when all three sides are known. The formula is as follows:
A = √(s(s-a)(s-b)(s-c))
where A is the area of the triangle, and s is the semi-perimeter, which is calculated as:
s = (a + b + c) / 2
Q: What is the semi-perimeter of a triangle?
A: The semi-perimeter of a triangle is half the sum of the lengths of its sides. It is calculated as:
s = (a + b + c) / 2
Q: How do I apply Heron's formula to find the area of a triangle?
A: To apply Heron's formula, follow these steps:
- Calculate the semi-perimeter of the triangle.
- Plug the semi-perimeter and the lengths of the sides into Heron's formula.
- Simplify the expression and calculate the square root.
Q: What are some common mistakes to avoid when using Heron's formula?
A: Some common mistakes to avoid when using Heron's formula include:
- Calculating the semi-perimeter incorrectly.
- Plugging in the wrong values into Heron's formula.
- Not simplifying the expression correctly.
- Not calculating the square root correctly.
Q: Can I use Heron's formula to find the area of any triangle?
A: Yes, you can use Heron's formula to find the area of any triangle, as long as you know the lengths of all three sides.
Q: What are some real-world applications of Heron's formula?
A: Heron's formula has many real-world applications, including:
- Architecture: Heron's formula is used to calculate the area of buildings and other structures.
- Engineering: Heron's formula is used to calculate the area of bridges and other infrastructure.
- Physics: Heron's formula is used to calculate the area of objects in motion.
Q: Can I use Heron's formula to find the area of a right triangle?
A: Yes, you can use Heron's formula to find the area of a right triangle. However, there is a simpler formula for finding the area of a right triangle, which is:
A = (base × height) / 2
Q: Can I use Heron's formula to find the area of an equilateral triangle?
A: Yes, you can use Heron's formula to find the area of an equilateral triangle. However, there is a simpler formula for finding the area of an equilateral triangle, which is:
A = (√3 / 4) × side²
Q: What are some tips for memorizing Heron's formula?
A: Some tips for memorizing Heron's formula include:
- Practice, practice, practice: The more you practice using Heron's formula, the more likely you are to remember it.
- Create a mnemonic device: Create a mnemonic device to help you remember the formula.
- Break the formula down into smaller parts: Break the formula down into smaller parts to make it easier to remember.
Q: Can I use Heron's formula to find the area of a triangle with negative side lengths?
A: No, you cannot use Heron's formula to find the area of a triangle with negative side lengths. The lengths of the sides of a triangle must be positive.
Q: Can I use Heron's formula to find the area of a triangle with zero side lengths?
A: No, you cannot use Heron's formula to find the area of a triangle with zero side lengths. The lengths of the sides of a triangle must be positive.
Conclusion
In conclusion, Heron's formula is a powerful tool for finding the area of a triangle. By following the steps outlined in this article, you should be able to use Heron's formula to find the area of any triangle. Remember to avoid common mistakes, practice using the formula, and break it down into smaller parts to make it easier to remember.