Contents
In this publication, we will consider the basic properties of height in an equilateral (regular) triangle. We will also analyze an example of solving a problem on this topic.
Note: the triangle is called equilateralif all its sides are equal.
Height properties in an equilateral triangle
Property 1
Any height in an equilateral triangle is both a bisector, a median, and a perpendicular bisector.
- BD – height lowered to the side AC;
- BD is the median that divides the side AC in half, i.e. AD = DC;
- BD – angle bisector ABC, i.e. ∠ABD = ∠CBD;
- BD is the median perpendicular to AC.
Property 2
All three altitudes in an equilateral triangle have the same length.
AE = BD = CF
Property 3
The heights in an equilateral triangle at the orthocenter (point of intersection) are divided in a ratio of 2:1, counting from the vertex from which they are drawn.
- AO = 2OE
- BO = 2OD
- CO = 2OF
Property 4
The orthocenter of an equilateral triangle is the center of the inscribed and circumscribed circles.
- R is the radius of the circumscribed circle;
- r is the radius of the inscribed circle;
- R = 2r (follows from Properties 3).
Property 5
The height in an equilateral triangle divides it into two equal-area (equal-area) right-angled triangles.
S1 = S2
Three heights in an equilateral triangle divide it into 6 right triangles of equal area.
Property 6
Knowing the length of the side of an equilateral triangle, its height can be calculated by the formula:
a is the side of the triangle.
Example of a problem
The radius of a circle circumscribed around an equilateral triangle is 7 cm. Find the side of this triangle.
Solution
As we know from properties 3 и 4, the radius of the circumscribed circle is 2/3 of the height of an equilateral triangle (h). Consequently, h = 7 ∶ 2 ⋅ 3 = 10,5 cm.
Now it remains to calculate the length of the side of the triangle (the expression is derived from the formula in Property 6):
