The first historians of our civilization - the ancient Greeks - mention Egypt as the birthplace of geometry. It is difficult to disagree with them, knowing with what stunning precision the giant tombs of the pharaohs were erected. The mutual arrangement of the planes of the pyramids, their proportions, orientation to the cardinal points - to achieve such perfection would be unthinkable without knowing the basics of geometry.
The very word "geometry" can be translated as "measurement of the earth." Moreover, the word "earth" does not appear as a planet - part Solar system, but as a plane. The marking of areas for farming, most likely, is the very original basis of the science of geometric shapes, their types and properties.
The triangle is the simplest spatial figure of the planimetry, containing only three points - the vertices (there is never less). The basis of the foundations, perhaps, is why something mysterious and ancient appears in him. The all-seeing eye inside the triangle is one of the earliest known occult signs, and the geography of its distribution and the time frame are simply amazing. From the ancient Egyptian, Sumerian, Aztec and other civilizations to more modern occult communities scattered around the globe.
What are triangles
Usual versatile triangle is closed geometric figure, consisting of three segments of different lengths and three angles, none of which is straight. In addition to him, there are several special types.
An acute-angled triangle has all angles less than 90 degrees. In other words, all the corners of such a triangle are sharp.
The right-angled triangle, over which at all times schoolchildren cried because of the abundance of theorems, has one angle with a magnitude of 90 degrees, or, as it is also called, a straight line.
An obtuse triangle differs in that one of its corners is obtuse, that is, its magnitude is more than 90 degrees.
An equilateral triangle has three sides of the same length. For such a figure, all angles are also equal.
And finally, in an isosceles triangle of three sides, two are equal.
Distinctive features
The properties of an isosceles triangle also determine its main, main difference - the equality of the two sides. These equal sides are usually called the hips (or, more often, the sides), but the third side is called the "base".
In the figure under consideration, a = b.
The second criterion for an isosceles triangle follows from the theorem of sines. Since sides a and b are equal, the sines of their opposite angles are also equal:
a / sin γ = b / sin α, whence we have: sin γ = sin α.
The equality of the sines implies the equality of the angles: γ = α.
So, the second sign of an isosceles triangle is the equality of the two angles adjacent to the base.
Third sign. In a triangle, elements such as height, bisector and median are distinguished.
If in the process of solving the problem it turns out that in the considered triangle any two of these elements coincide: height with bisector; bisector with median; median with height - we can definitely conclude that the triangle is isosceles.
Geometric properties of the figure
1. Properties of an isosceles triangle. One of the distinguishing qualities of the figure is the equality of the angles adjacent to the base:
<ВАС = <ВСА.
2. One more property was considered above: the median, bisector and height in an isosceles triangle coincide if they are built from its top to the base.
3. Equality of bisectors drawn from the vertices at the base:
If AE is the bisector of the angle BAC, and CD is the bisector of the angle BCA, then: AE = DC.
4. The properties of an isosceles triangle also provide for the equality of heights, which are drawn from the vertices at the base.
If we construct the heights of the triangle ABC (where AB = BC) from the vertices A and C, then the obtained segments CD and AE will be equal.
5. Equal will also be the medians drawn from the corners at the base.
So, if AE and DC are medians, that is, AD = DB, and BE = EC, then AE = DC.
Height of an isosceles triangle
The equality of the sides and angles at them introduces some peculiarities in the calculation of the lengths of the elements of the figure in question.
The height in an isosceles triangle divides the figure into 2 symmetrical right-angled triangles, the sides of which protrude with the hypotenuses. The height in this case is determined according to the Pythagorean theorem, like a leg.
A triangle can have all three sides equal, then it will be called equilateral. The height in an equilateral triangle is determined in the same way, only for calculations it is enough to know only one value - the length of the side of this triangle.
You can determine the height in another way, for example, knowing the base and the angle adjacent to it.
Median of an isosceles triangle
The considered type of triangle, due to its geometric features, is solved quite simply by the minimum set of initial data. Since the median in an isosceles triangle is equal to both its height and its bisector, the algorithm for its determination is no different from the order in which these elements are calculated.
For example, you can determine the length of the median by the known lateral side and the value of the apex angle.
How to determine the perimeter
Since the two sides of the considered planimetric figure are always equal, it is enough to know the length of the base and the length of one of the sides to determine the perimeter.
Consider an example when you need to determine the perimeter of a triangle from a known base and height.
The perimeter is equal to the sum of the base and twice the length of the side. The lateral side, in turn, is defined using the Pythagorean theorem as the hypotenuse of a right triangle. Its length is equal to the square root of the sum of the square of the height and the square of half of the base.
Area of an isosceles triangle
As a rule, it is not difficult to calculate the area of an isosceles triangle. The universal rule for determining the area of a triangle as half the product of the base and its height applies, of course, in our case. However, the properties of an isosceles triangle make the task easier again.
Let us assume that the height and angle adjacent to the base are known. It is necessary to determine the area of the figure. You can do it this way.
Since the sum of the angles of any triangle is 180 °, it is not difficult to determine the value of the angle. Next, using the proportion made up according to the theorem of sines, the length of the base of the triangle is determined. Everything, the base and the height - sufficient data to determine the area - are available.
Other properties of an isosceles triangle
The position of the center of a circle circumscribed around an isosceles triangle depends on the magnitude of the apex angle. So, if an isosceles triangle is acute-angled, the center of the circle is located inside the figure.
The center of a circle that is circumscribed around an obtuse isosceles triangle lies outside it. And finally, if the angle at the apex is 90 °, the center lies exactly in the middle of the base, and the diameter of the circle passes through the base itself.
In order to determine the radius of a circle circumscribed about an isosceles triangle, it is sufficient to divide the length of the lateral side by twice the cosine of half the value of the apex angle.
The properties of an isosceles triangle express the following theorems.
Theorem 1. In an isosceles triangle, the angles at the base are equal.
Theorem 2. In an isosceles triangle, the bisector to the base is the median and the height.
Theorem 3. In an isosceles triangle, the median drawn to the base is the bisector and the height.
Theorem 4. In an isosceles triangle, the height drawn to the base is the bisector and the median.
Let us prove one of them, for example, Theorem 2.5.
Proof. Consider an isosceles triangle ABC with base BC and prove that ∠ B = ∠ C. Let AD be the bisector of triangle ABC (Fig. 1). Triangles ABD and ACD are equal by the first sign of equality of triangles (AB = AC by condition, AD is a common side, ∠ 1 = ∠ 2, since AD is a bisector). It follows from the equality of these triangles that ∠ B = ∠ C. The theorem is proved.
Using Theorem 1, the following theorem is established.
Theorem 5. The third criterion for the equality of triangles. If three sides of one triangle are respectively equal to three sides of another triangle, then such triangles are equal (Fig. 2).
Comment. The sentences set forth in examples 1 and 2 express the properties of the midpoint perpendicular to the line segment. It follows from these sentences that the mid-perpendiculars to the sides of the triangle intersect at one point.
Example 1. Prove that the point of the plane equidistant from the ends of the segment lies on the perpendicular to this segment.
Solution. Let point M be equidistant from the ends of the segment AB (Fig. 3), ie, AM = BM.
Then Δ AMB is isosceles. Let us draw a straight line p through point M and the middle O of segment AB. The segment MO by construction is the median of the isosceles triangle AMB, and therefore (Theorem 3), and the height, that is, the straight line MO, is the median perpendicular to the segment AB.
Example 2. Prove that each point of the perpendicular to the segment is equidistant from its ends.
Solution. Let p be the midpoint perpendicular to the segment AB and point O - the midpoint of the segment AB (see Fig. 3).
Consider an arbitrary point M lying on the line p. Let's draw the segments AM and VM. Triangles AOM and PTO are equal, since they have straight angles at apex O, leg OM is common, and leg OA is equal to leg OB by condition. From the equality of the triangles AOM and PTO it follows that AM = BM.
Example 3. In triangle ABC (see Fig. 4) AB = 10 cm, BC = 9 cm, AC = 7 cm; in a triangle DEF DE = 7 cm, EF = 10 cm, FD = 9 cm.
Compare triangles ABC and DEF. Find correspondingly equal angles.
Solution. These triangles are equal in the third attribute. Accordingly, equal angles: A and E (lie opposite the equal sides BC and FD), B and F (lie opposite the equal sides AC and DE), C and D (lie opposite the equal sides AB and EF).
Example 4. In Figure 5 AB = DC, BC = AD, ∠B = 100 °.
Find Angle D.
Solution. Consider triangles ABC and ADC. They are equal in the third criterion (AB = DC, BC = AD by condition and the AC side is common). From the equality of these triangles it follows that ∠ В = ∠ D, but the angle В is equal to 100 °, which means that the angle D is equal to 100 °.
Example 5. In an isosceles triangle ABC with base AC, the outer angle at apex C is 123 °. Find the angle ABC. Give your answer in degrees.
Video solution.
Among all triangles, there are two special types: right-angled triangles and isosceles triangles. Why are these types of triangles so special? Well, firstly, such triangles very often turn out to be the main characters of the USE tasks in the first part. And secondly, problems about right-angled and isosceles triangles are much easier to solve than other problems in geometry. You just need to know a few rules and properties. All the most interesting is discussed in the corresponding topic, but now we will consider isosceles triangles. And above all, what is an isosceles triangle. Or, as mathematicians say, what is the definition of an isosceles triangle?
See how it looks:
Like a right-angled triangle, an isosceles triangle has special names for its sides. Two equal sides are called lateral sides and the third party is basis.
And again, pay attention to the picture:
It may, of course, be like this:
So be careful: side - one of two equal sides in an isosceles triangle, and the base is a third party.
Why is an isosceles triangle so good? To understand this, let's draw the height to the base. Do you remember what height is?
So what happened? From one isosceles triangle, two rectangular ones turned out.
This is already good, but it will turn out this way in any, the most "coosbral" triangle.
What is the difference between the picture for an isosceles triangle? Look again:
Well, first of all, of course, it is not enough for these strange mathematicians to simply see - they must certainly prove. And then suddenly these triangles are slightly different, and we will consider them the same.
But don't worry: in this case, proving is almost as easy as seeing.
Let's start? Look carefully, we have:
And that means! Why? Yes, we just find and, and from the Pythagorean theorem (remembering at the same time that)
Have you made sure? Well, now we have
And on three sides - the easiest (third) sign of the equality of triangles.
Well, our isosceles triangle has split into two identical rectangular ones.
See how interesting it is? It turned out that:
How is it customary to talk about this among mathematicians? Let's go in order:
(Remember here that the median is the line drawn from the vertex that divides the side in half, and the bisector is the angle.)
Well, here we discussed what good can be seen if given an isosceles triangle. We deduced that the angles at the base of an isosceles triangle are equal, and the height, bisector and median, drawn to the base, coincide.
And now another question arises: how to recognize an isosceles triangle? That is, as mathematicians say, what are signs of an isosceles triangle?
And it turns out that you just need to "turn" all statements on the contrary. This, of course, is not always the case, but an isosceles triangle is still a great thing! What happens after the "overturn"?
Well, look:
If the height and the median coincide, then:
If the height and the bisector coincide, then: 
If the bisector and median coincide, then:
Well, don't forget and use:
- If you are given an isosceles triangular triangle, feel free to draw the height, get two right triangles and solve the problem about a right triangle.
- If given that two angles are equal then triangle exactly isosceles and you can hold the height and .... (The house that Jack built ...).
- If it turns out that the height is halved on the side, then the triangle is isosceles with all the ensuing bonuses.
- If it turns out that the height has divided the angle into the floors - isosceles too!
- If the bisector divided the side in half or the median is the angle, then this also happens only in an isosceles triangle
Let's see how it looks in tasks.
Problem 1(the simplest)
In a triangle, the sides and are equal, and. Find.
We decide:
First a drawing.
What is the foundation here? Certainly, .
We remember that if, then and.
Updated drawing:
Let us denote by. What is the sum of the angles of the triangle there? ?
We use:
That's answer: .
Not difficult, right? Even the height was not necessary.
Task 2(Also not very tricky, but you need to repeat the topic)
In a triangle,. Find.
We decide:
The triangle is isosceles! We draw the height (this is the trick with the help of which everything will be solved now).
Now we "delete from life", we will only consider.
So, in we have:
Remembering the tabular values of cosines (well, or looking at the cheat sheet ...)
It remains to find:.
Answer: .
Note that we have here very required knowledge concerning a right-angled triangle and "tabular" sines and cosines. Very often it happens: topics, "isosceles triangle" and in puzzles go in bundles, but with other topics they are not very friendly.
Isosceles triangle. Average level.
These two equal sides are called lateral sides, a the third side is the base of an isosceles triangle.
Look at the picture: and - the sides, - the base of the isosceles triangle.
Let's understand in one picture why this is so. Let's draw the height from the point.
This means that they have equal all the corresponding elements.
Everything! In one fell swoop (height) they proved all the statements at once.
And remember: to solve the isosceles triangle problem, it is often very useful to lower the height to the base of the isosceles triangle and divide it into two equal right-angled triangles.
Signs of an isosceles triangle
The converse statements are also true:
Almost all of these statements can again be proved "in one fell swoop."
1. So, let in were equal to and.
Let's draw the height. Then
2.a) Now let in some triangle height and bisector match.
2.b) And if the height and the median coincide? Everything is almost the same, no more complicated!
 |
- on two legs |
2.c) But if there is no height, which is lowered to the base of an isosceles triangle, then there are no initially right-angled triangles. Badly!
But there is a way out - read it in the next level of theory, because here the proof is more complicated, but for now, just remember that if the median and bisector coincide, then the triangle will also be isosceles, and the height will still coincide with this bisector and median.
Let's summarize:
- If the triangle is isosceles, then the angles at the base are equal, and the height, bisector and median drawn to the base coincide.
- If in some triangle there are two equal angles, or some two of the three lines (bisector, median, height) coincide, then such a triangle is isosceles.
Isosceles triangle. Brief description and basic formulas
An isosceles triangle is a triangle that has two equal sides.
Signs of an isosceles triangle:
- If in some triangle two angles are equal, then it is isosceles.
- If in some triangle coincide:
a) height and bisector or
b) height and median or
v) median and bisector,
drawn to one side, then such a triangle is isosceles.
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