Geometry/Triangle

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A triangle is a type of polygon having three sides and, therefore, three angles. The triangle is a closed figure formed from three straight line segments joined at their ends. The points at the ends can be called the corners, angles, or vertices of the triangle. Since any given triangle lies completely within a plane, triangles are often treated as two-dimensional geometric figures. As such, a triangle has no volume and, because it is a two-dimensionally closed figure, the flat part of the plane inside the triangle has an area, typically referred to as the area of the triangle. Triangles are always convex polygons.

A triangle must have at least some area, so all three corner points of a triangle cannot lie in the same line. The sum of the lengths of any two sides of a triangle is always greater than the length of the third side. The preceding statement is sometimes called the Triangle Inequality.

Certain types of triangles[edit]

Categorized by angle[edit]

The sum of the interior angles in a triangle always equals 180o. This means that no more than one of the angles can be 90o or more. All three angles can all be less than 90oin the triangle; then it is called an acute triangle. One of the angles can be 90o and the other two less than 90o; then the triangle is called a right triangle. Finally, one of the angles can be more than 90o and the other two less; then the triangle is called an obtuse triangle.

Categorized by sides[edit]

If all three of the sides of a triangle are of different length, then the triangle is called a scalene triangle.

If two of the sides of a triangle are of equal length, then it is called an isoceles triangle. In an isoceles triangle, the angle between the two equal sides can be more than, equal to, or less than 90o. The other two angles are both less than 90o.

If all three sides of a triangle are of equal length, then it is called an equilateral triangle and all three of the interior angles must be 60o, making it equilangular. Because the interior angles are all equal, all equilateral triangles are also the three-sided variety of a regular polygon and they are all similar, but might not be congruent. However, polygons having four or more equal sides might not have equal interior angles, might not be regular polygons, and might not be similar or congruent. Of course, pairs of triangles which are not equilateral might be similar or congruent.

Further discussion of Congruent Triangles and Similar Triangles may be found in those corresponding sections.

Opposite corners and sides in triangles[edit]

Picture of Triangle ABC

If one of the sides of a triangle is chosen, the interior angles of the corners at the side's endpoints can be called adjacent angles. The corner which is not one of these endpoints can be called the corner opposite to the side. The interior angle whose vertex is the opposite corner can be called the angle opposite to the side.

Likewise, if a corner or its angle is chosen, then the two sides sharing an endpoint at that corner can be called adjacent sides. The side not having this corner as one of its two endpoints can be called the side opposite to the corner.

The sides or their lengths of a triangle are typically labeled with lower case letters. The corners or their corresponding angles can be labeled with capital letters. The triangle as a whole can be labeled by a small triangle symbol and its corner points. In a triangle, the largest interior angle is opposite to longest side, and vice versa.

Any triangle can be divided into two right triangles by taking the longest side as a base, and extending a line segment from the opposite corner to a point on the base such that it is perpendicular to the base. Such a line segment would be considered the height or altitude ( h ) for that particular base ( b ). The two right triangles resulting from this division would both share the height as one of its sides. The interior angles at the meeting of the height and base would be 90o for each new right triangle. For acute triangles, any of the three sides can act as the base and have a corresponding height. For more information on right triangles, see Right Triangles and Pythagorean Theorem.

Area of Triangles[edit]

Triangle (acute) bh.PNG

If base and height of a triangle are known, then the area of the triangle can be calculated by the formula:

S = \frac{b \times h}{2}

(S is the symbol for area)


Ways of calculating the area inside of a triangle are further discussed under Area.

Centres[edit]

The centroid is constructed by drawing all the medians of the triangle. All three medians intersect at the same point: this crossing point is the centroid. Centroids are always inside a triangle. They are also the centre of gravity of the triangle.

The three angle bisectors of the triangle intersect at a single point, called the incentre. Incentres are always inside the triangle. The three sides are equidistant from the incentre. The incentre is also the centre of the inscribed circle (incircle) of a triangle, or the interior circle which touches all three sides of the triangle.

The circumcentre is the intersection of all three perpendicular bisectors. Unlike the incentre, it is outside the triangle if the triangle is obtuse. Acute triangles always have circumcentres inside, while the circumcentre of a right triangle is the midpoint of the hypotenuse. The vertices of the triangle are equidistant from the circumcentre. The circumcentre is so called because it is the centre of the circumcircle, or the exterior circle which touches all three vertices of the triangle.

The orthocentre is the crossing point of the three altitudes. It is always inside acute triangles, outside obtuse triangles, and on the right vertex of the right-angled triangle.

Please note that the centres of an equilateral triangle are always the same point.