Geometry is an extremely interesting science, which they begin to teach in Russian schools in the seventh grade. But sometimes the topic covered in the lesson is not at all clear, and attempts to read a paragraph in the textbook only exacerbate the situation. Then the omniscient Internet comes to the rescue or some students simply open their finished homework, which is fundamentally wrong, because then the question remains unanswered, the brain does not develop, there are even greater problems with the perception of information in the lesson, which leads to poor grades. In this article we will analyze one of the basic elements with which many problems are solved. What is the definition of the height of the triangle? How to build it? You will find answers to these and many other questions in this article.
Determining the height of a triangle
Understanding the essence of the element, and why it is needed, always begins with the study of theory. So, the height of the triangle is a perpendicular, lowered from the top of the triangle to a straight line containing the opposite side. Why not on the side itself? We will deal with this a bit later.
How many heights can be drawn in a triangle? The number of heights coincides with the number of vertices, that is, three. All three intersections of the perpendiculars of the triangle intersect at one point.
Let's also repeat the theory of two other important elements - the bisector and the median.
A bisector is a beam connecting the top of a triangle to the opposite side, while dividing the angle into two equal parts.
Median - a segment connecting the top of the corner with the middle of the opposite side.
Types of Triangles
There are a lot of varieties of triangles in geometry, in each of them heights play a role. Let's look in detail all the types of this figure. Determining the height of the triangle will help us with this.
Let's start with the usual acute-angled versatile triangle, in which all angles are sharp and not equal to 60 degrees, and the sides are not equal to each other. In this geometric figure, the heights intersect, but this point will not be the center of the triangle.
In an obtuse triangle, the degree measure of one angle is greater than 90 degrees. The height coming out of an obtuse angle is lowered to a straight line containing the opposite side.
The next is an isosceles triangle. He has only two sides and two corners at the base. Interestingly, the height drawn from the top to the base of the triangle coincides with the median and bisector.
In an equilateral triangle, all sides and angles are equal, which are 60 degrees (each). All heights, medians and bisectors coincide and intersect at one point - the center of the triangle.
Standard height related formulas
For each of the above cases, there are formulas for determining the height, but in this paragraph we will consider only those that are suitable for each type of triangle. There are four such formulas.
- The simplest and most affordable: H = 2S / a. Knowing the area and length of the side to which the perpendicular is lowered, we can find the height by dividing the double product of the area by the side.
- If the triangle is enclosed in a circle, then there is a formula for this case: H = bc / 2R. To find the height, you need to divide the sides on which the perpendicular does not fall, divided by the double product of the radius of the circle circumscribed around the triangle.
- Knowing only the sides, we can also find the height: H = (2√ (p (pa) * (pb) * (pc))) / a, where: p is the half-perimeter; a - side on which the height is lowered; b, c - parties on which the perpendicular does not fall.
- And for those who know, it has already begun to undergo trigonometry and knows what sine and cosine are, there is such a formula: H = bsinY = csinB. Sinus is the ratio of the opposite side to the perpendicular; H is the perpendicular; b and c are the sides opposite the angles Y and B, respectively.
Right triangle
You might think that we forgot about right triangles, but that’s not so. A right-angled triangle is a triangle in which one of the angles is 90 degrees. There is only one height in a right-angled triangle, because the other two are sides, or rather legs. A single perpendicular comes out of a right angle and descends to the hypotenuse. There are a lot of formulas for finding for this case:
- H = ab / c;
- H = ab / √ (a 2 + b 2 );
- H = csinAcosA = c sinBcosB;
- H = bsinA = a sinB;
- H = √de.
Where:
H is the height;
a, b - legs;
c - hypotenuse;
A, B - angles during hypotenuse;
d, e - segments obtained from dividing the hypotenuse by height.
Conclusion
So, in this article we looked at determining the height of a triangle. What are the types of triangles? What formulas can be used to find the height? Now you can give detailed, and most importantly correct answers to all these questions.