Equilateral

Definition

An equilateral shape, such as a triangle or polygon, is one in which all sides are of equal length. This unique property gives equilateral shapes distinct geometric characteristics that are important in the study of rectangles, triangles, and trapezoids.

5 Must Know Facts For Your Next Test

1. An equilateral triangle has all three sides of equal length and all three interior angles of equal measure (60 degrees).
2. The diagonals of an equilateral rectangle intersect at right angles and are of equal length.
3. Equilateral trapezoids have two parallel sides of equal length and two non-parallel sides of equal length.
4. The sum of the interior angles in any equilateral polygon is given by the formula: (n-2) * 180°, where n is the number of sides.
5. Equilateral shapes exhibit high degrees of symmetry, with lines of symmetry passing through the midpoints of each side and the vertices.

Review Questions

• How do the properties of an equilateral triangle relate to the properties of rectangles and trapezoids?
  • The equal side lengths and angles of an equilateral triangle are key features that connect it to the properties of other geometric shapes. For example, the 60-degree interior angles of an equilateral triangle are also found in the non-parallel sides of an equilateral trapezoid. Additionally, the equal diagonal lengths and right-angle intersections of an equilateral rectangle are related to the symmetry and congruence properties of equilateral triangles.
• Explain how the formula for the sum of interior angles in a polygon relates to the properties of an equilateral polygon.
  • The formula for the sum of interior angles in a polygon, (n-2) * 180°, where n is the number of sides, is particularly relevant for equilateral polygons. Since all the interior angles in an equilateral polygon are equal, this formula can be used to determine the measure of each individual angle. For example, in an equilateral triangle (n=3), the formula gives (3-2) * 180° = 180°, meaning each interior angle is 60 degrees. This relationship between the number of sides and the interior angle measure is a key property of equilateral polygons.
• Analyze how the symmetry properties of an equilateral shape, such as a triangle or rectangle, influence its overall geometric characteristics.
  • The high degree of symmetry exhibited by equilateral shapes, such as the lines of symmetry passing through the midpoints of sides and vertices, is a defining feature that greatly influences their geometric properties. This symmetry ensures that all sides and angles are congruent, leading to unique characteristics like equal diagonals in equilateral rectangles and the 60-degree interior angles in equilateral triangles. The symmetry also contributes to the regular polygon nature of equilateral shapes, where all sides and angles are equal. This symmetry is a fundamental aspect of equilateral shapes that underpins many of their important geometric relationships within the context of rectangles, triangles, and trapezoids.

Related terms

Equiangular: A shape where all interior angles are equal in measure, such as an equilateral triangle.

Regular Polygon: A polygon where all sides are equal in length and all interior angles are equal in measure.

Congruent:

Two shapes or figures that are exactly the same size and shape, with all corresponding parts being equal.