You triangles have remarkable points with many applications.. Some of these elements, such as height, median, bisector and bisector, which are given by straight segments inside the triangle, they have important characteristics and applications, not only in mathematics.
We know that the intersection of two or more straight lines is given by a point, so the meeting of these segments form points that have important characteristics and properties, they are:
- orthocenter
- barycenter
- circumcenter
- center
triangle height
the height of a triangle is the segment formed by the union of one of the vertices with its opposite side or its extension, in which a 90° angle is formed between the segment and the side. In every triangle, it is possible to draw three relative heights to each side. Look:
the segment AG is the height relative to side BC, and the segment DH is the height relative to the EF side. Note that in order to determine the height relative to the EF side, it was necessary to carry out an extension of the side.
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Orthocenter
The orthocenter is the intersection of the heights relative to the three vertices, that is, it is meeting point between all heights of a triangle.
The point O is the orthocenter of triangle ABC.
The orthocenter has some important properties in some types of triangle, see:
→ No acute triangle, the heights and orthocenter are inside the figure.
→ In one right triangle, two heights are coincident with the two sides, another height is inside the triangle, and the orthocenter is located at the vertex of that triangle, which has an angle of 90°.
→ In one obtuse triangle, one of the heights is inside the triangle, and the other two are outside it, the orthocenter is also located on this outside.
Read too: Triangle classifications: criteria and names
median
The median of a triangle is the segment formed by the union of one of its vertices with the midpoint of the side opposite that vertex. Note that, in a triangle, it is possible to determine three medians relative to each side, see:
The line segment CD is the median relative to side AB. Note that this segment has split side AB into two equal parts, that is, in half.
Barycenter
The barycenter is given by the intersection of the three medians of a triangle, that is, by the meeting point of the three medians, see:
The point G is the center of triangle ABC.
As in the orthocenter, the barycenter has some important properties, see:
→ The barycenter will determine in each of the median segments that satisfy each of the equalities.
Example 1
Knowing that point G in the following image is the barycenter of triangle ABC and that GD = 3 cm, determine the length of segment CG.
From the barycenter properties, we know that the ratio between the GD and CG segment is equal to one-half. So, replacing these values in the relationship, we have:
→ Considering the definition of median, see that all medians are inside the triangle, so we can conclude that the barycenter of any triangle is also always inside the figure.. This observation is valid for any triangle.
The barycenter also gives us an important physical characteristic of triangles, as it allows us to balance them, that is, the barycenter is the center of mass of a triangle.
See too: Sine, cosine, tangent - trigonometric ratios
Mediatrix
The bisector of a triangle is given by a perpendicular line that passes through the midpoint on one side of this triangle.
Circumcenter
The circumcenter is defined by the meeting of the bisectors, that is, by the intersection between them. If we represent a triangle inscribed in a circumference, we will see that the circumcenter is the center of this circumference, see:
The point Mis the circumcenter of triangle ABC and the center of the circumference. Points H, I and J are, respectively, the midpoints of sides CB, CA and AB.
The circumcenter also has some properties when drawn on the right-angled triangle, obtuse-angle, and acute-angle.
→ The circumcenter in the right triangle is the midpoint of the hypotenuse.
→ The circumcenter in a obtuse triangle is on the outside.
→ The circumcenter in a acute triangle it stays inside.
Also access: Circle and circumference – what are the differences?
Bisector
The bisector of a triangle is given by the straight line that divides an internal angle of the triangle. When drawing the internal bisector, see that we will have three internal bisectors relative to the three sides of the triangle:
center
The center is given by intersection of the internal bisectors of a triangle, that is, it is given by the meeting of these semi-straights. Since the bisectors are internal, the incenter will always be inside the triangle as well.
Incentro has some useful properties to solve some problems, see some of them:
→ The center of a circle inscribed in a triangle coincides with the incenter of that figure.
→ The incenter of a triangle is equidistant from all its sides, that is, the distances between the incenter and the three sides of the triangle are all equal.
solved exercises
question 1 – Knowing that the segment in the interior is the bisector relative to the side AC and that the measurements shown in the figure represent the angle divided by the bisector, determine the value of x.
Resolution
By the definition of a bisector, we know that it divides the internal angle of a triangle in half, that is, into two equal parts, so we have to:
5x -10 = 3x + 20
solving the first degree equation, we'll have to:
5x – 10 = 3x + 20
5x - 3x = 20 + 10
2x = 30
x = 15
Therefore, x = 15.
question 2 – The perpendicular line segment drawn from a vertex of a triangle to one of its sides is called:
the height
b) bisector
c) bisector
d) median
e) base
Resolution
From the definitions we studied, we saw that the only one that satisfies the utterance condition is height. Remember that height is the segment perpendicular to one side of a triangle.
by Robson Luiz
Maths teacher
