Question
Is this statement true or false?
In a regular polygon, two consecutive radii and their included side form an equilateral
triangle.
false
In a regular polygon, two consecutive radii and their included side form an equilateral
triangle.
false
Ask by Stewart Guzman. in the United States
Jan 23,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The statement is false.
Solution
To determine whether the statement is true or false, let’s analyze it step by step.
-
Understanding the terms:
- A regular polygon is a polygon with all sides and angles equal.
- A radius of a polygon refers to the distance from the center of the polygon to any of its vertices.
- The included side refers to the side of the polygon that is between two consecutive radii.
-
Visualizing the scenario:
- In a regular polygon, if we take two consecutive vertices and draw the radii to these vertices, we form a triangle with the included side (the side of the polygon connecting these two vertices).
-
Analyzing the triangle:
- The two radii are equal in length (since they are both radii of the same circle).
- The included side is the side of the polygon, which may or may not be equal to the lengths of the radii.
-
Conditions for an equilateral triangle:
- For the triangle formed by the two radii and the included side to be equilateral, all three sides must be equal.
- This is only true in specific cases, such as in an equilateral triangle itself or in a regular polygon with a specific number of sides (like a triangle).
-
Conclusion:
- In general, for a regular polygon with more than three sides, the included side will not equal the lengths of the radii, thus not forming an equilateral triangle.
Therefore, the statement is false.
Answered by UpStudy AI and reviewed by a Professional Tutor
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Simplify this solution The Deep Dive
In a regular polygon, the claim is indeed false. An equilateral triangle requires all three sides to be of equal length, which only happens in a regular polygon when the number of sides is three! In any regular polygon with more than three sides, the two consecutive radii will be equal, but the included side will be shorter, creating a triangle that is isosceles rather than equilateral.
For example, in a regular pentagon, the angles formed at the center and the side length don’t allow for an equilateral formation. You can visualize this better by sketching out a pentagon with its radii; you’ll see the angles are just too sharp to maintain equality among all three sides. Keep drawing those polygons to develop an instinct for their unique properties!
