Problem 163 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Find the in-radius of the triangle having sides \(13,14,15\).

Step-by-Step Solution

Answer

The in-radius of the triangle with sides 13, 14, and 15 is approximately \(2.18\).

Step 1: Calculate the semi-perimeter of the triangle

Using the formula \(s = (a + b + c) / 2\), and substituting the given lengths of the sides into this formula, we get \(s = (13 + 14 + 15) / 2 = 21\)

Step 2: Calculate the in-radius of the triangle

Now using the formula for the in-radius \(r = \sqrt{s(s-a)(s-b)(s-c)} / s\), substituting the calculated semi-perimeter and the lengths of the sides into this formula, we get \(r = \sqrt{21 * (21 - 13) * (21 - 14) * (21 - 15)} / 21\) This simplifies to: \(r = \sqrt{21 * 8 * 7 * 6} / 21\)

Step 3: Simplify the expression

Simplify the expression \(r = \sqrt{21 * 8 * 7 * 6} / 21\) and get the final value of the in-radius. After simplifying, we found that \(r = \sqrt{12348} / 21 \approx 2.18\)

Key Concepts

Semi-Perimeter Formula

The semi-perimeter formula is a crucial concept in geometry, especially when working with triangles. It provides a way to find half of a triangle's perimeter. By knowing the lengths of the sides of the triangle, we can apply this formula to aid in calculations involving certain triangle properties, such as the inradius.

To calculate the semi-perimeter, you add up all the side lengths of the triangle and divide the sum by two. For example, with a triangle having sides of lengths 13, 14, and 15, the semi-perimeter would be calculated as follows:

• Add the side lengths: 13 + 14 + 15 = 42
• Divide the sum by 2: 42/2 = 21

So, the semi-perimeter, denoted as \( s \), is 21. This result plays a key role in subsequent calculations, such as determining the inradius of the triangle.

Triangle Side Lengths

The side lengths of a triangle are fundamental to finding various triangle attributes. Here, the side lengths are given as 13, 14, and 15. From this information, we can not only find the semi-perimeter as discussed earlier, but also go further to calculate other properties such as the area and the inradius.

When working with these side lengths in specific formulas:

• Ensure each side is correctly substituted into the equations.
• Remain consistent with labeling sides as \( a \), \( b \), and \( c \), which helps in avoiding confusion.

In our example, we label the side lengths as \( a = 13 \), \( b = 14 \), and \( c = 15 \). These values need precision during calculations to maintain accuracy. By using these lengths with the semi-perimeter, more detailed mathematical properties like the inradius can be explored.

Simplification of Expressions

Simplification of expressions is a vital skill when calculating geometric properties such as the inradius. Once the semi-perimeter \( s \) is determined, and side lengths are known, we can substitute these values into the formula for the inradius of a triangle:\[ r = \frac{\sqrt{s(s-a)(s-b)(s-c)}}{s}\]After inputting the values, calculations often result in complex expressions. For example, in our case:\[ r = \frac{\sqrt{21 \times 8 \times 7 \times 6}}{21}\]The next step is to simplify this expression by multiplying and reducing it as much as possible:

• Calculate inside: 21 \times 8 \times 7 \times 6 = 7056
• Then take the square root: \( \sqrt{7056} \approx 84 \)
• Finally, divide by 21, simplifying to \( r \approx 4 \)

The final result shows the inradius is approximately 4, illustrating how managing and simplifying expressions leads to practical and easily interpretable results.