Question
Find the area of a circle
inscribed is a right-angle
triangle (leave your anower
in pi) witt length
inscribed is a right-angle
triangle (leave your anower
in pi) witt length
Ask by Ball Bush. in Ghana
Mar 20,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The area of the inscribed circle is
Solution
-
Compute the semiperimeter
of the triangle: -
Calculate the area
of the right triangle. Since the triangle is right-angled, we can use the legs and as the base and height: -
Find the inradius
of the inscribed circle using the formula: -
Compute the area of the inscribed circle:
Thus, the area of the circle is
Answered by UpStudy AI and reviewed by a Professional Tutor
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Simplify this solution Mind Expander
To find the area of a circle inscribed in a right triangle, you first need to find the radius of the inscribed circle (inradius). For a right triangle, the inradius
where
Plugging in the values:
The area
So, the area of the inscribed circle is
The relationship between triangles and circles goes way back to ancient civilizations. The Greeks, particularly in the works of Euclid, explored circle properties to study triangles, including inscribed circles. This beautiful connection between shapes reveals how geometry was perceived within the realm of mathematics, demonstrating that even in antiquity, the complexity of these figures was appreciated and analyzed!
Understanding inscribed circles has practical implications, especially in architecture and engineering, where maximizing space and aesthetics is key. When designing structures that incorporate circular elements, knowledge of inscribed shapes helps designers determine optimal dimensions and ensure a perfect fit within existing frameworks, leading to both functional and visually appealing designs!
