The shortest distance from the incenter to the sides of a triangle is equal to:
A. the radius of the inscribed circle.
B. the diameter of the inscribed circle.
C. the circumference of the inscribed circle.
D. a tangent of the inscribed circle.
Answer (2)
The incenter is the center of the inscribed circle.
The inscribed circle is tangent to each side of the triangle.
The shortest distance from the incenter to a side is the radius of the inscribed circle.
Therefore, the answer is: the radius of the inscribed circle.
Explanation
Understanding the Incenter and Inscribed Circle The incenter of a triangle is the center of its inscribed circle. The inscribed circle is tangent to each side of the triangle. The shortest distance from the incenter to any side is the radius of the inscribed circle, as the radius is perpendicular to the tangent at the point of tangency.
Determining the Shortest Distance The shortest distance from the incenter to the sides of a triangle is the radius of the inscribed circle.
Selecting the Correct Option Comparing the shortest distance to the given options:
A. the radius of the inscribed circle.
B. the diameter of the inscribed circle.
C. the circumference of the inscribed circle.
D. a tangent of the inscribed circle.
The correct answer is A.
Final Answer The shortest distance from the incenter to the sides of a triangle is equal to the radius of the inscribed circle.
Examples
Imagine you're designing a triangular park with a circular pond at its center (the incenter). The shortest path from the pond's edge to any side of the park is the radius of the pond. This ensures that every side of the park is equally accessible from the pond, making it a balanced and user-friendly design.
The shortest distance from the incenter to the sides of a triangle is equal to the radius of the inscribed circle. This is because the radius is the perpendicular distance from the incenter to the tangent points on the triangle's sides. Therefore, the chosen option is A: the radius of the inscribed circle.
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