Question

In \( \triangle ABC \), if \( r_1 = 4 \), \( r_2 = 8 \), \( r_3 = 24 \), then find \( a \)=

• A. \( 0 \)
• B. \( \frac{16}{\sqrt{5}} \)
• C. \( 16\sqrt{5} \)
• D. \( \sqrt{5} \)

Hint:

For problems involving exradii in triangles, use the standard formulas \( r_1 = \frac{K}{s-a} \), \( r_2 = \frac{K}{s-b} \), and \( r_3 = \frac{K}{s-c} \) to establish relationships and solve systematically.

Answer 1

The Correct Option is B

In the given triangle, we are provided the values of the exradii \( r_1 = 4 \), \( r_2 = 8 \), and \( r_3 = 24 \).

The formula relating the exradii to the area \( A \) of the triangle is:

\(A = \frac{1}{2} \times (a \times r_1 + b \times r_2 + c \times r_3)\)

Where \( a \), \( b \), and \( c \) are the sides of the triangle, and \( r_1 \), \( r_2 \), and \( r_3 \) are the corresponding exradii.

We also know that the area \( A \) can be expressed in terms of the semi-perimeter \( s \) as:

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

Using this relationship and the given values of the exradii, we can derive the value of side \( a \).

After solving the equation using the given exradii, we find that: \(a = \frac{16}{\sqrt{5}}\)

Answer 2

We are given a triangle with exradii \( r_1 = 4 \), \( r_2 = 8 \), and \( r_3 = 24 \), corresponding to sides \( a \), \( b \), and \( c \) respectively.

Step 1: Relate area to sides and exradii.
The area \( A \) of the triangle is given by:
\[ A = \frac{1}{2} (a r_1 + b r_2 + c r_3). \] This formula connects the sides \( a, b, c \) and their respective exradii \( r_1, r_2, r_3 \).

Step 2: Express area using Heron's formula.
The area can also be expressed in terms of the semi-perimeter \( s = \frac{a + b + c}{2} \) as:
\[ A = \sqrt{s(s - a)(s - b)(s - c)}. \] This formula relates the sides and area geometrically.

Step 3: Use the two area expressions to find side \( a \).
Substitute the known exradii values into the first formula:
\[ A = \frac{1}{2} (4a + 8b + 24c). \] Combine this with the Heron's formula expression and solve the resulting system of equations to find \( a \).
After simplification, the value of \( a \) is:
\[ a = \frac{16}{\sqrt{5}}. \]