How to find the height of a triangle?

To solve many geometric problems it is required to find the height of a given figure. These tasks are of practical importance. When carrying out construction work, determining the height helps to calculate the required amount of materials, as well as to determine how accurately the slopes and openings are made. Often, to create patterns, you need to have an idea of the properties of geometric shapes.

Many people, despite good grades at school, when constructing ordinary geometric figures, raise the question of how to find the height of a triangle or parallelogram. And the definition of the height of the triangle is the most difficult. This is because the triangle can be sharp, blunt, isosceles or rectangular. For each of the types of triangles, there are rules of construction and calculation.

How to find the height of a triangle in which all angles are sharp, graphically

If all the angles of the triangle are sharp (each angle in the triangle is less than 90 degrees), then to find the height you need to do the following.

1. Given the given parameters, we construct a triangle.
2. We introduce the notation. A, B and C will be the vertices of the figure. The angles corresponding to each vertex are α, β, γ. The sides opposing these angles are a, b, c.
3. The height is the perpendicular dropped from the vertex of the corner to the opposite side of the triangle. To find the heights of a triangle, we construct perpendiculars: from the vertex of the angle α to the side a, from the vertex of the angle β to the side b, and so on.
4. The point of intersection of height and side a is denoted by H1, and the height h1. The point of intersection of height and side b is H2, the height is h2, respectively. For the side c, the height is h3, and the intersection point is H3.

Next, for each kind of triangle, we use the same notation for the sides, angles, heights and vertices of the triangles.

Height in triangle with obtuse angle

Now consider how to find the height of a triangle if one corner is blunt (more than 90 degrees). In this case, the height drawn from the obtuse angle will be inside the triangle. The other two heights will be outside the triangle.

Suppose that in our triangle the angles α and β are acute, and the angle γ is obtuse. Then, to construct the heights emerging from the angles α and β, we must continue the opposite sides of the triangle in order to draw perpendiculars.

How to find the height of an isosceles triangle

Such a figure has two equal sides and a base, while the angles at the base are also equal. This equality of sides and angles facilitates the construction of heights and their calculation.

First, draw the triangle itself. Let the sides b and c, as well as the angles β, γ be respectively equal.

Now draw the height from the vertex of the angle α, denote it by h1. For an isosceles triangle, this height will be both a bisector and a median.

Next, we construct two other heights: h2 for side b and angle β, h3 for side c and angle γ. These heights will be equal in length.

For the base, you can do only one construction. For example, to hold the median - a segment that connects the vertex of an isosceles triangle and the opposite side, the base, to find the height and the bisector. And to calculate the height length for the other two sides, you can only build one height. Thus, in order to graphically determine how to calculate the height of an isosceles triangle, it is sufficient to find two heights of three.

How to find the height of a right triangle

In a rectangular triangle, it is much easier to determine the heights than others. This is because the cateches themselves constitute a right angle, which means they are heights.

To build the third height, as usual, a perpendicular is drawn connecting the vertex of the right angle and the opposite side. As a result, in order to learn how to find the height of a triangle in this case, only one construction is required.