How to find the height in an isosceles triangle? The finding formula, the height properties in an isosceles triangle

Geometry is not only an object in the school, on which you need to get an excellent assessment. It is also knowledge that is often required in life. For example, when building a house with a high roof, you need to calculate the thickness of the logs and their number. This is easy if you know how to find the height in an isosceles triangle. Architectural structures are based on knowledge of the properties of geometric shapes. Forms of buildings often visually resemble them. Egyptian pyramids, packages with milk, artistic embroidery, northern murals and even patties are all triangles surrounding a person. As Plato said, the whole world is based on triangles.

Isosceles triangle

To make it clearer, what will be discussed next, it is worth remembering the basics of geometry.

A triangle is isosceles if it has two equal sides. They are always called lateral. The side, the dimensions of which differ, was called the grounds.

Basic concepts

Like any science, geometry has its own basic rules and concepts. There are a lot of them. Consider only those without which our topic will be somewhat incomprehensible.

Height is a straight line drawn perpendicular to the opposite side.

The median is a segment directed from any vertex of the triangle exclusively to the middle of the opposite side.

The angle bisector is a ray that divides the angle in half.

The triangle bisector is a straight line, or rather, a segment of the bisector of the corner, connecting the vertex with the opposite side.

It is very important to remember that the angle bisector is necessarily a ray, and the bisector of a triangle is part of such a ray.

Angles at the base

The theorem says that the angles located at the base of any isosceles triangle are always equal. It is very simple to prove this theorem. Consider the isosceles triangle ABC shown, for which AB = BC. From the angle ABC it is necessary to draw a bisector of the VD. Now consider the two triangles obtained. By the condition AB = BC, the side of the AP for the triangles is common, and the angles of the AVD and SVD are equal, because the VA is a bisector. Recalling the first sign of equality, we can safely conclude that the triangles under consideration are equal. And consequently, all corresponding angles are equal. And, of course, the parties, but by this point we will return later.

The height of an isosceles triangle

The main theorem on which the solution of almost all problems is based is as follows: the height in an isosceles triangle is a bisector and a median. To understand its practical meaning (or essence), it is necessary to make an auxiliary allowance. For this it is necessary to cut out an isosceles triangle from paper. The easiest way to do this is from a standard tetrad sheet in the cell.

Fold the resulting triangle in half, aligning the sides. What happened? Two equal triangles. Now you need to check the guesswork. Unfold the origami. Draw a line of fold. Using a protractor, check the angle between the drawn line and the base of the triangle. What does the angle of 90 degrees say? The fact that the line drawn is a perpendicular. By definition - height. How to find the height in an isosceles triangle, we sorted it out. Now let's deal with the corners at the top. Using the same protractor, check the angles formed now by the height. They are equal. Hence, the height is also a bisector. Armed with a ruler, measure the lengths on which the height of the base breaks. They are equal. Consequently, the height in an isosceles triangle divides the base in half and is a median.

Proof of the theorem

The visual aid clearly demonstrates the truth of the theorem. But geometry - science is quite accurate, therefore it requires proof.

During the consideration of the equality of angles at the base, the equality of triangles was proved. Recall VD is a bisector, and the triangles of AED and SVD are equal. The conclusion was this: the corresponding sides of the triangle and, of course, the angles are equal. Hence, AD = SD. Therefore, VD is the median. It remains to prove that VD is a height. Proceeding from the equality of the triangles under consideration, it turns out that the angle of the ADB is equal to the angle of the VDV. But these two corners are contiguous, and, as is known, give a total of 180 degrees. Therefore, what are they equal to? Of course, 90 degrees. Thus, VD is the height in an isosceles triangle drawn to the base. Q.E.D.

Main features

• To successfully solve problems, one should remember the basic features of isosceles triangles. They seem to be inverse to the theorems.
• If in the course of solving a problem the equality of two angles is found, then you are dealing with an isosceles triangle.
• If it was possible to prove that the median is simultaneously the height of the triangle, boldly conclude that the triangle is isosceles.
• If the bisector is also the height, then, based on the main features, the triangle is referred to as isosceles.
• And, of course, if the median appears in the role of height, then such a triangle is isosceles.

Height formula 1

However, for most problems it is required to find the arithmetic height. That is why we consider how to find the height in an isosceles triangle.

Let us return to the above figure ABC, in which a - sides, in - base. VD is the height of this triangle, it has the designation h.

What is the triangle of AED? Since VD is the height, the triangle of the AVD is rectangular, the cathet of which is to be found. Using the formula of Pythagoras, we get:

AV² = АД² + ВД²

Having determined from the expression VD and substituting the previously used notation, we obtain:

Н² = а² - (в / 2) ².

It is necessary to extract the root:

H = √a² - in² / 4.

If we take out from the sign of the root ¼, then the formula will look like:

H = ½ √4a² - in².

This is the height in an isosceles triangle. The formula follows from the theorem of Pythagoras. Even if you forget this symbolic entry, then, knowing the method of finding, you can always withdraw it.

Height formula 2

The formula described above is the main one and is most often used in solving most geometric problems. But she is not the only one. Sometimes in the condition, instead of the base, the value of the angle is given. With such data, how to find the height in an isosceles triangle? To solve similar problems it is advisable to use a different formula:

H = a / sin α,

Where H is the height directed to the base,

But - the side,

Α is the angle at the base.

If the task gives the value of the angle at the vertex, then the height in an isosceles triangle is as follows:

H = a / cos (β / 2),

Where H is the height dropped on the base,

Β is the angle at the vertex,

And - the side.

Rectangular isosceles triangle

A very interesting property is the triangle, whose vertex is 90 degrees. Consider a right triangle ABC. As in the previous cases, VD is the height directed to the base.

The angles at the base are equal. Calculate their great work will not be:

Α = (180-90) / 2.

Thus, the angles at the base are always 45 degrees. Now consider the triangle ADV. It is also rectangular. Let us find the angle of the ABD. By simple calculations, we get 45 degrees. And, consequently, this triangle is not only rectangular, but also isosceles. The parties AD and VD are lateral sides and are equal among themselves.

But the BP side at the same time is half of the AC side. It turns out that the height in an isosceles triangle is half the base, and if written in the form of a formula, we get the following expression:

H = B / 2.

It should be remembered that this formula is an exclusively particular case, and can only be used for rectangular isosceles triangles.

Golden triangles

Very interesting is the golden triangle. In this figure, the ratio of the lateral to the base equals the value called the Phidias number. The angle at the top is 36 degrees, at the base - 72 degrees. This triangle was admired by the Pythagoreans. The principles of the golden triangle are the basis of many immortal masterpieces. Known to all five-pointed star is built on the intersection of isosceles triangles. For many creations, Leonardo da Vinci used the principle of the "golden triangle". Composition "Gioconda" is based precisely on the figures that create a regular star pentagon.

The picture "Cubism", one of the creations of Pablo Picasso, fascinates the view put in the basis of an isosceles triangle.