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Geometry: Triangles





 



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In geometry as well as other math, a triangle is a geometric figure in a plane that has three sides. There are many ways of classifying triangles. One way is by the type of angles that they contain, a triangle that does not have any angle greater than or equal to 90° is called an acute triangle. A triangle with a 90° is called a right triangle. If the triangle has an angle of greater than 90° then it is called an obtuse triangle. In an equiangular triangle, the measure of the angles are all equal.

An acute triangle has angles that are all less than 90 degrees.A right triangle has one right angle of 90 degrees and two acute angles less than 90 degrees.An obtuse angle has one angle greater than 90 degrees and two acute angles less than 90 degrees.
acute triangleright triangleobtuse triangle






Triangle Definition

A triangle can be defined as the union of three line segments joined in such a way that the endpoints of the line segments are not collinear and the triangle has an interior region. The line segments are referred to as sides while the endpoints joined together are called vertices. Each point of the triangle is called a vertex. It has three angles which add to 180°.

The interior region of the triangle is referred to as area and the sum of the lengths of its line segments or sides is called its perimeter. The symbol used to denote a triangle is .

For example, ABC can be represented as follows:

Triangles can be identified by their labeled vertices. Each vertex is labeled by a letter.





Another way of describing a triangle is by whether a point is in the interior or the exterior of the angles. A point is in the interior of the triangle if it is in the interior of all the angles of the triangle.

The same goes for the exterior. A point is in the exterior of a triangle if it is in the exterior of the angles. Another way of classifying triangles is by the length of their sides.

If the length of all the sides are the same then it is called an equilateral triangle. If two of the sides have the same length then it is called an isosceles triangle.

If none of the sides are equal, it is called a scalene triangle.







Equilateral triangles have equal length on all sides and the interior angles are all 60 degrees.

equilateral triangle







Isosceles triangles have equal length on two sides of the triangle.

isosceles triangle







The sides of a scalene triangle have unequal lengths.

scalene triangle







It should also be noted that an equilateral triangle is the same as an equiangular triangle.

One important property of the triangle is called the Triangle Inequality.

Triangle Inequality
The length of the side of a triangle is always less than the sum of the lengths of the other two sides.



Triangle inequality is where the sum of the lengths of two sides of a triangle are always greater than the length of the third side.

triangle inequality with lengths a b c



If a is the length of side 2, b is the length of side 3 and c is the length of side 4 then:

a < b + c

          2 < 3 + 4

b < a + c

          3 < 2 + 4

c < a + b

          4 < 2 + 3





Sometimes it is necessary to describe certain parts of a triangle. For example, if one side of a triangle is horizontal when looking at it, it is usually referred to as its base. A perpendicular line can be drawn from the base to the top vertex which is then called the altitude or height.

The altitude of a triangle is also its height and is perpendicular to the base.

triangle with base and altitude





The median is a line segment drawn from a vertex to the midpoint on the opposite side. The median also acts as a bisector for the angle at the vertex as well as for the line segment opposite the angle and vertex.

The median of a triangle is a line segment connected from the vertex of the triangle to the midpoint on the opposite side and is also an angle bisector.

triangle with median





Triangle Area

The area, A, of a triangle can be calculated in one of three ways:

1)    A = ½ * Base * Height
or
A = ½ bh
The area of a triangle is one half times base times height.

base and height for triangle area



A = ½ * 4 * 2 = 4

The area is 4 units.





The area of a triangle is one half times base times height.

base and height for triangle area



A = ½ * 5 * 8 = 20

The area is 20 units.





2)     Heron’s Formula
Let s be equal to half the perimeter:
s = ½(a+b+c)
then:
A = √s(s-a)(s-b)(s-c)




Heron's formula can also be used to compute the area of a triangle.

perimeter for triangle area



s = ½ * ( 7 + 12 + 17 ) = 18

A = √18(18-7)(18-12)(18-17) = √18(11)(6)(1) = 34.5

The area is 34.5 units.





3)    Triangle with inscribed circle
A = ½ rP
P = a + b + c
The area of a triangle is one half times radius times the perimeter.

radius and perimeter for triangle area



P = 6 + 14 + 18 = 38

A = ½ * 2 * 38 = 38

The area is 38 units.





Equilateral Triangle

Some triangles have special properties such as the equilateral triangle, the isosceles triangle and the right triangle.

Equilateral Triangle Corollary
Each Angle in an equilateral triangle measures 60°




An equilateral triangle measures 60 degrees at all interior angles.

The equilateral triangle has all congruent angles





Right Triangles

The longest side of a right triangle that is opposite of the right angle is called the hypotenuse while the other two sides are called legs.

One of the properties of a right triangle is that the Pythagorean theorem can be applied to it.

Pythagorean Theorem
The square of the hypotenuse is equal to the sum of the squares of the legs.

It is also written as:

     c2 = a2 + b2
or
c = √ a2 + b2

In the isosceles right triangle, the legs are equal to each other and the two angles other than the right angle are equal to 45°. Also, the hypotenuse is equal to √2 times one of the legs: h = √2s.

An isosceles right triangle has angles that measure 45, 45 and 90 degrees.

isosceles right triangle





 





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