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: \(a = \frac{16}{\sqrt{5}}\)