Definition and properties of the height of a triangle

Contents

Determining the Height of a Triangle
Height in different types of triangles
Triangle Height Properties

In this publication, we will consider determining the height of a triangle, demonstrate how it looks depending on the type of triangle, and also list its main properties.

Content

Determining the Height of a Triangle

Height of triangle – this is a perpendicular that is lowered from the top of the figure to the opposite side.

height base – a point on the opposite side of the triangle, which is crossed by the height (or the point of intersection of their extensions).

Height is usually denoted by the letter h (sometimes like h_a – this means that it is held to the side a).

Height in different types of triangles

Depending on the type of figure, the height can be:

pass inside the triangle (in acute-angled △);
pass outside the triangle (in an obtuse △);
be one of the legs (in a rectangular △), except for the height drawn to the hypotenuse.

Triangle Height Properties

Property 1

All three heights in a triangle (or their extensions) intersect at one point, which is called orthocenter (dot O in the drawings below).

in an acute triangle;
in an obtuse triangle;
in a right triangle.
Vertex A is, among other things, the point of intersection of heights.

Property 2

When two heights intersect in a triangle, the following similar triangles are formed:

△ABE∼△CBF: two corners (∠ABC – general, ∠General Conditions of Purchase and ∠SFBC are straight).
△AFG∼△CEG: two corners (∠AFG and ∠CEG – straight lines, ∠AGF and ∠CGE equal as vertical angles).
△ABC∼△BEF: three equal angles (∠ABC = ∠EBF, ∠ACB = ∠SFOE, ∠CAB = ∠BEF).
Note: the proof of the similarity of the last pair of triangles is quite long and is not the purpose of this article, so we will dwell on it in detail.