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tam giác cân sửa
Isosceles triangle
Triangle with at least two sides congruent
In geometry, an isosceles triangle is a triangle that has two sides of equal length. Sometimes it is specified as having exactly two sides of equal length, and sometimes as having at least two sides of equal length, the latter version thus including the equilateral triangle as a special case. Examples of isosceles triangles include the isosceles right triangle, the golden triangle, and the faces of bipyramids and certain Catalan solids.
"Isosceles" redirects here. For other uses, see Isosceles (disambiguation).
Quick Facts Type, Edges and vertices ...
The mathematical study of isosceles triangles dates back to ancient Egyptian mathematics and Babylonian mathematics. Isosceles triangles have been used as decoration from even earlier times, and appear frequently in architecture and design, for instance in the pediments and gables of buildings.
The two equal sides are called the legs and the third side is called the base of the triangle. The other dimensions of the triangle, such as its height, area, and perimeter, can be calculated by simple formulas from the lengths of the legs and base. Every isosceles triangle has an axis of symmetry along the perpendicular bisector of its base. The two angles opposite the legs are equal and are always acute, so the classification of the triangle as acute, right, or obtuse depends only on the angle between its two legs.
Terminology, classification, and examples
Euclid defined an isosceles triangle as a triangle with exactly two equal sides, but modern treatments prefer to define isosceles triangles as having at least two equal sides. The difference between these two definitions is that the modern version makes equilateral triangles (with three equal sides) a special case of isosceles triangles. A triangle that is not isosceles (having three unequal sides) is called scalene. "Isosceles" is made from the Greek roots "isos" (equal) and "skelos" (leg). The same word is used, for instance, for isosceles trapezoids, trapezoids with two equal sides, and for isosceles sets, sets of points every three of which form an isosceles triangle.
In an isosceles triangle that has exactly two equal sides, the equal sides are called legs and the third side is called the base. The angle included by the legs is called the vertex angle and the angles that have the base as one of their sides are called the base angles. The vertex opposite the base is called the apex. In the equilateral triangle case, since all sides are equal, any side can be called the base.
Special isosceles triangles
Isosceles right triangle
Three congruent inscribed squares in the Calabi triangle
A golden triangle subdivided into a smaller golden triangle and golden gnomon
The triakis triangular tiling
Catalan solids with isosceles triangle faces
Triakis tetrahedron
Triakis octahedron
Tetrakis hexahedron
Pentakis dodecahedron
Triakis icosahedron
Whether an isosceles triangle is acute, right or obtuse depends only on the angle at its apex. In Euclidean geometry, the base angles can not be obtuse (greater than 90°) or right (equal to 90°) because their measures would sum to at least 180°, the total of all angles in any Euclidean triangle. Since a triangle is obtuse or right if and only if one of its angles is obtuse or right, respectively, an isosceles triangle is obtuse, right or acute if and only if its apex angle is respectively obtuse, right or acute. In Edwin Abbott's book Flatland, this classification of shapes was used as a satire of social hierarchy: isosceles triangles represented the working class, with acute isosceles triangles higher in the hierarchy than right or obtuse isosceles triangles.
As well as the isosceles right triangle, several other specific shapes of isosceles triangles have been studied. These include the Calabi triangle (a triangle with three congruent inscribed squares), the golden triangle and golden gnomon (two isosceles triangles whose sides and base are in the golden ratio), the 80-80-20 triangle appearing in the Langley's Adventitious Angles puzzle, and the 30-30-120 triangle of the triakis triangular tiling. Five Catalan solids, the triakis tetrahedron, triakis octahedron, tetrakis hexahedron, pentakis dodecahedron, and triakis icosahedron, each have isosceles-triangle faces, as do infinitely many pyramids and bipyramids.
Formulas
Height
For any isosceles triangle, the following six line segments coincide:
the altitude, a line segment from the apex perpendicular to the base,the angle bisector from the apex to the base,the median from the apex to the midpoint of the base,the perpendicular bisector of the base within the triangle,the segment within the triangle of the unique axis of symmetry of the triangle, andthe segment within the triangle of the Euler line of the triangle, except when the triangle is equilateral.
Their common length is the height {\displaystyle h} of the triangle. If the triangle has equal sides of length {\displaystyle a} and base of length {\displaystyle b}, the general triangle formulas for the lengths of these segments all simplify to
{\displaystyle h={\sqrt {a^{2}-{\frac {b^{2}}{4}}}}.}
This formula can also be derived from the Pythagorean theorem using the fact that the altitude bisects the base and partitions the isosceles triangle into two congruent right triangles.
The Euler line of any triangle goes through the triangle's Koyeuduong (thảo luận) 01:50, ngày 18 tháng 1 năm 2022 (UTC)