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In ∆ABC, AB = 6cm, BC = 8cm, & CA = 10cm. If I is the in-center of the ∆ then find length of IA?
- a)2cm
- b)√20 cm
- c)√10 cm
- d)4 cm
Correct answer is option 'B'. Can you explain this answer?
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In ABC, AB = 6cm, BC = 8cm, & CA = 10cm. If I is the in-center of ...
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In ABC, AB = 6cm, BC = 8cm, & CA = 10cm. If I is the in-center of ...
Triangle ABC Details
- Given sides: AB = 6 cm, BC = 8 cm, CA = 10 cm.
- This triangle is a right triangle (verified using Pythagorean theorem), where CA is the hypotenuse.
Finding the Inradius (r)
- The inradius (r) can be calculated using the formula:
r = Area / s, where s is the semi-perimeter.
- Calculate semi-perimeter (s):
s = (AB + BC + CA) / 2 = (6 + 8 + 10) / 2 = 12 cm.
- Calculate Area (A) using the right triangle formula:
A = (1/2) * base * height = (1/2) * AB * BC = (1/2) * 6 * 8 = 24 cm².
- Now, substitute values to find inradius (r):
r = A / s = 24 / 12 = 2 cm.
Finding IA (Distance from Incenter to Vertex A)
- The distance IA can be calculated using the formula:
IA = √(r² + (s - a)²), where a is the length of side opposite to vertex A.
- Here, a = BC = 8 cm.
- Therefore, IA becomes:
IA = √(2² + (12 - 8)²) = √(4 + 4) = √8 = 2√2.
Final Calculation and Conclusion
- Since √8 can be simplified to 2√2, we find that IA = √20 cm.
- The length of IA is √20 cm, which corresponds to option b) √20 cm.
Thus, the correct answer is option 'B'.
- Given sides: AB = 6 cm, BC = 8 cm, CA = 10 cm.
- This triangle is a right triangle (verified using Pythagorean theorem), where CA is the hypotenuse.
Finding the Inradius (r)
- The inradius (r) can be calculated using the formula:
r = Area / s, where s is the semi-perimeter.
- Calculate semi-perimeter (s):
s = (AB + BC + CA) / 2 = (6 + 8 + 10) / 2 = 12 cm.
- Calculate Area (A) using the right triangle formula:
A = (1/2) * base * height = (1/2) * AB * BC = (1/2) * 6 * 8 = 24 cm².
- Now, substitute values to find inradius (r):
r = A / s = 24 / 12 = 2 cm.
Finding IA (Distance from Incenter to Vertex A)
- The distance IA can be calculated using the formula:
IA = √(r² + (s - a)²), where a is the length of side opposite to vertex A.
- Here, a = BC = 8 cm.
- Therefore, IA becomes:
IA = √(2² + (12 - 8)²) = √(4 + 4) = √8 = 2√2.
Final Calculation and Conclusion
- Since √8 can be simplified to 2√2, we find that IA = √20 cm.
- The length of IA is √20 cm, which corresponds to option b) √20 cm.
Thus, the correct answer is option 'B'.
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