Problem 99 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Find the area of the triangle having sides 13,14 and \(15 \mathrm{~cm}\).

Step-by-Step Solution

Answer

The area of the triangle with sides 13 cm, 14 cm, and 15 cm is 84 square centimeters.

Step 1: Calculate the semi-perimeter

The semi-perimeter, \(s\), can be calculated using the formula: \(s = \frac{a+b+c}{2}\) Substitute the given values for the sides a = 13 cm, b = 14 cm, and c = 15 cm: \(s = \frac{13 + 14 + 15}{2} = \frac{42}{2} = 21\) Step 2: Apply Heron's formula

Step 2: Apply Heron's formula

Heron's formula is: Area = \(\sqrt{s(s-a)(s-b)(s-c)}\) Substitute the values we have calculated: Area = \(\sqrt{21(21 - 13)(21 - 14)(21 - 15)}\) Area = \(\sqrt{21(8)(7)(6)}\) Step 3: Calculate the area

Step 3: Calculate the area

Use the provided values to compute the area: Area = \(\sqrt{21(8)(7)(6)} = \sqrt{21168} = 84\) The area of the triangle is 84 square centimeters.

Key Concepts

Understanding the Semi-Perimeter

Calculating the semi-perimeter is an important step when using Heron's formula to find the area of a triangle. The semi-perimeter, denoted as \( s \), is essentially half of the perimeter of a triangle.

To compute the semi-perimeter, you add up the lengths of all three sides of the triangle, and then divide that sum by two. The formula looks like this:

• \( s = \frac{a + b + c}{2} \)

Where \( a \), \( b \), and \( c \) are the lengths of the sides of the triangle.

For example, with a triangle that has sides of 13 cm, 14 cm, and 15 cm, you would calculate:

• \( s = \frac{13 + 14 + 15}{2} = 21 \)

This value is crucial as it forms part of the Heron's formula, a powerful tool for calculating more complex triangle areas without needing the height.

Triangle Area Calculation using Heron's Formula

Heron's formula is a beautiful method for finding the area of a triangle when you know the lengths of all three sides. It does not require the height of the triangle, making it especially useful when the height is unknown or difficult to determine.

The formula is expressed as:

• Area = \( \sqrt{s(s-a)(s-b)(s-c)} \)

Here, \( s \) is the semi-perimeter, and \( a \), \( b \), and \( c \) are the side lengths. Using the semi-perimeter calculated in the previous step, you can substitute the values into the formula to find the area.

So, for a triangle with sides 13 cm, 14 cm, and 15 cm, the area is calculated as follows:

• Area = \( \sqrt{21(21-13)(21-14)(21-15)} \)
• Area = \( \sqrt{21 \times 8 \times 7 \times 6} \)
• Area = \( \sqrt{21168} = 84 \) square centimeters

This tells us the area of the triangle is 84 square centimeters, illustrating the power of Heron's formula.

Enhancing Mathematics Problem-Solving Skills

Problem-solving in mathematics often involves breaking down complex problems into more manageable parts, and using formulas like Heron's requires several logical steps.

To effectively utilize Heron's formula, you should:

• Comprehend the problem and identify the given data, such as side lengths.
• Calculate intermediary steps, like the semi-perimeter, which can bring clarity and ensure accuracy.
• Apply the formula carefully, ensuring each mathematical operation is executed correctly.

This methodical approach ensures no step is overlooked, validating the procedure and boosting confidence in the solution. As students practice using Heron's formula and similar techniques, they enhance their overall problem-solving abilities by developing skills in:

• Logical reasoning – carefully interpreting problem statements and translating them into mathematical operations.
• Analytical thinking – following a structured method to dissect and solve equations.
• Attention to detail – ensuring precision in calculations to avoid errors.

By refining these skills, students are better equipped to tackle not only geometry problems but a wide array of mathematical challenges.