Question:
In \( \triangle ABC \), if \( a = 30, b = 24, c = 18 \), then \( r_3 \) is equal to
In \( \triangle ABC \), if \( a = 30, b = 24, c = 18 \), then \( r_3 \) is equal to
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To calculate the excircle radius, use Heron's formula to find the area and the semi-perimeter to compute the required excircle radius.
Updated On: Apr 22, 2026
- 15
- 18
- 36
- 12
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The Correct Option is D
Solution and Explanation
Step 1: Understand the problem and the given information.
We are given a triangle \( \triangle ABC \) with sides \( a = 30 \), \( b = 24 \), and \( c = 18 \).
We are asked to find \( r_3 \), which refers to the radius of the excircle opposite vertex \( C \).
Step 2: Use the formula for the excircle radius.
The radius \( r_3 \) of the excircle opposite vertex \( C \) is given by the formula: \[ r_3 = \frac{K}{s - c} \] where \( K \) is the area of the triangle and \( s \) is the semi-perimeter.
Step 3: Calculate the semi-perimeter.
The semi-perimeter \( s \) is given by: \[ s = \frac{a + b + c}{2} = \frac{30 + 24 + 18}{2} = 36 \]
Step 4: Calculate the area using Heron's formula.
The area \( K \) of the triangle can be calculated using Heron's formula: \[ K = \sqrt{s(s - a)(s - b)(s - c)} \] Substituting the values of \( s \), \( a \), \( b \), and \( c \): \[ K = \sqrt{36(36 - 30)(36 - 24)(36 - 18)} = \sqrt{36 \times 6 \times 12 \times 18} \] Simplifying: \[ K = \sqrt{36 \times 6 \times 12 \times 18} = \sqrt{46656} = 216 \]
Step 5: Substitute into the formula for \( r_3 \).
Now we can substitute the values of \( K \), \( s \), and \( c \) into the formula for \( r_3 \): \[ r_3 = \frac{216}{36 - 18} = \frac{216}{18} = 12 \]
Step 6: Conclusion.
Thus, the radius of the excircle opposite vertex \( C \) is \( r_3 = 12 \), corresponding to option (D).
We are given a triangle \( \triangle ABC \) with sides \( a = 30 \), \( b = 24 \), and \( c = 18 \).
We are asked to find \( r_3 \), which refers to the radius of the excircle opposite vertex \( C \).
Step 2: Use the formula for the excircle radius.
The radius \( r_3 \) of the excircle opposite vertex \( C \) is given by the formula: \[ r_3 = \frac{K}{s - c} \] where \( K \) is the area of the triangle and \( s \) is the semi-perimeter.
Step 3: Calculate the semi-perimeter.
The semi-perimeter \( s \) is given by: \[ s = \frac{a + b + c}{2} = \frac{30 + 24 + 18}{2} = 36 \]
Step 4: Calculate the area using Heron's formula.
The area \( K \) of the triangle can be calculated using Heron's formula: \[ K = \sqrt{s(s - a)(s - b)(s - c)} \] Substituting the values of \( s \), \( a \), \( b \), and \( c \): \[ K = \sqrt{36(36 - 30)(36 - 24)(36 - 18)} = \sqrt{36 \times 6 \times 12 \times 18} \] Simplifying: \[ K = \sqrt{36 \times 6 \times 12 \times 18} = \sqrt{46656} = 216 \]
Step 5: Substitute into the formula for \( r_3 \).
Now we can substitute the values of \( K \), \( s \), and \( c \) into the formula for \( r_3 \): \[ r_3 = \frac{216}{36 - 18} = \frac{216}{18} = 12 \]
Step 6: Conclusion.
Thus, the radius of the excircle opposite vertex \( C \) is \( r_3 = 12 \), corresponding to option (D).
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