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 determine how accurately the slopes and openings are made. Often, to build 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, the question arises of how to find the height of a triangle or parallelogram. Moreover, determining the height of the triangle is the most difficult. This is because the triangle can be sharp, blunt, isosceles or rectangular. Each type of triangle has its own rules for constructing and calculating.
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, do the following.
- According to the given parameters, we construct a triangle.
- 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. Opposing these corners are a, b, c.
- Height is the perpendicular, lowered from the top of the corner to the opposite side of the triangle. To find the heights of the triangle, we construct perpendiculars: from the vertex of the angle α to side a, from the top of the angle β to side b, and so on.
- The point of intersection of the height and side a is denoted by H1, and the height h1 itself. The intersection point of the height and side b will be H2, the height, respectively, h2. For side c, the height will be h3, and the intersection point H3.
Further, for each type of triangle we will use the same notation for the sides, angles, heights and vertices of the triangles.
Height in obtuse triangle
Now consider how to find the height of a triangle if one corner is obtuse (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 sharp, and the angle γ is obtuse. Then, to build the heights coming out of the angles α and β, it is necessary to continue the opposite sides of the triangle 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 to each other. 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 a 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 foundation, only one construction can be done. For example, draw a median - a segment connecting the top of an isosceles triangle and the opposite side, the base, to find the height and bisector. And to calculate the height length for the other two sides, you can build only one height. Thus, in order to graphically determine how to calculate the height of an isosceles triangle, it is enough to find two out of three heights.
How to find the height of a right triangle
It is much easier to determine the heights of a right-angled triangle than others. This is because the legs themselves constitute a right angle, which means they are heights.
To build a third height, as usual, a perpendicular is drawn connecting the top of the right angle and the opposite side. As a result, in order to find out how to find the height of the triangle in this case, only one construction is required.