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 amazing precision the giant tombs of the pharaohs were erected. Mutual arrangement 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 word “geometry” itself 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. Marking areas for maintenance Agriculture, most likely, is the very original basis of the science of geometric figures, their types and properties.
A triangle is the simplest spatial figure of planimetry, containing only three points - vertices (there are no fewer). The basis of the foundations, perhaps that is why something mysterious and ancient seems to be in him. The all-seeing eye inside a triangle is one of the earliest known occult signs, and the geography of its distribution and time frame are simply amazing. From ancient Egyptian, Sumerian, Aztec and other civilizations to more modern communities of occult lovers scattered across the globe.
What are triangles?
An ordinary scalene triangle is a closed geometric figure, consisting of three segments of different lengths and three angles, none of which are straight. In addition to this, there are several special types.
An acute triangle has all angles less than 90 degrees. In other words, all the angles of such a triangle are acute.
A right triangle, over which schoolchildren have always cried because of the abundance of theorems, has one angle of 90 degrees or, as it is also called, a straight line.
An obtuse triangle is distinguished by the fact that one of its angles is obtuse, that is, its size is more than 90 degrees.
An equilateral triangle has three sides of equal length. In such a figure, all angles are also equal.
And finally, at isosceles triangle Of the three sides, two are equal to each other.
Distinctive features
The properties of an isosceles triangle also determine its main, main difference - the equality of its two sides. These equal sides are usually called the hips (or, more often, the sides), and 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 equal:
a/sin γ = b/sin α, whence we have: sin γ = sin α.
From the equality of sines follows the equality of angles: γ = α.
So, the second sign of an isosceles triangle is the equality of two angles adjacent to the base.
Third sign. In a triangle, there are such elements as altitude, bisector and median.
If, in the process of solving the problem, it turns out that in the triangle in question any two of these elements coincide: the height with the bisector; bisector with median; median with height - we can definitely conclude that the triangle is isosceles.
Geometric properties of a figure
1. Properties of an isosceles triangle. One of the distinctive qualities of the figure is the equality of the angles adjacent to the base:
<ВАС = <ВСА.
2. One more property was discussed above: the median, bisector and altitude in an isosceles triangle coincide if they are built from its vertex to its base.
3. Equality of bisectors drawn from the vertices at the base:
If AE is the bisector of angle BAC, and CD is the bisector of angle BCA, then: AE = DC.
4. The properties of an isosceles triangle also provide for the equality of the heights that are drawn from the vertices at the base.
If we construct the altitudes of triangle ABC (where AB = BC) from vertices A and C, then the resulting segments CD and AE will be equal.
5. The medians drawn from the corners at the base will also be equal.
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 with them introduces some features into the calculation of the lengths of the elements of the figure under consideration.
The altitude in an isosceles triangle divides the figure into 2 symmetrical right triangles, the hypotenuses of which are on the sides. The height in this case is determined according to the Pythagorean theorem as 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 a similar 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, by knowing the base and the angle adjacent to it.
Median of an isosceles triangle
The type of triangle under consideration, due to its geometric features, can be solved quite simply using a minimal set of initial data. Since the median in an isosceles triangle is equal to both its height and its bisector, the algorithm for determining it is no different from the procedure for calculating these elements.
For example, you can determine the length of the median by the known lateral side and the magnitude of the apex angle.
How to determine the perimeter
Since the two sides of the planimetric figure under consideration are always equal, to determine the perimeter it is enough to know the length of the base and the length of one of the sides.
Let's consider an example when you need to determine the perimeter of a triangle using 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 the base.
Area of an isosceles triangle
As a rule, calculating the area of an isosceles triangle does not cause difficulties. The universal rule for determining the area of a triangle as half the product of the base and its height is applicable, of course, in our case. However, the properties of an isosceles triangle again make the task easier.
Let us assume that the height and angle adjacent to the base are known. It is necessary to determine the area of the figure. This can be done this way.
Since the sum of the angles of any triangle is 180°, it is not difficult to determine the size of the angle. Next, using the proportion compiled according to the theorem of sines, the length of the base of the triangle is determined. Everything, base and 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 vertex angle. So, if an isosceles triangle is acute, the center of the circle is located inside the figure.
The center of a circle circumscribed around an obtuse isosceles triangle lies outside it. And finally, if the angle at the vertex 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 enough to divide the length of the side by twice the cosine of half the vertex angle.
Among all triangles, there are two special types: right triangles and isosceles triangles. Why are these types of triangles so special? Well, firstly, such triangles extremely often turn out to be the main characters in the problems of the Unified State Exam in the first part. And secondly, problems about right and isosceles triangles are much easier to solve than other geometry problems. You just need to know a few rules and properties. All the most interesting things are discussed in the corresponding topic, but now let’s look at isosceles triangles. And first of all, what is an isosceles triangle? Or, as mathematicians say, what is the definition of an isosceles triangle?
See what it looks like:
Like a right triangle, an isosceles triangle has special names for its sides. Two equal sides are called sides, and the third party - basis.
And again pay attention to the picture:
It could, of course, be like this:
So be careful: lateral side - one of two equal sides in an isosceles triangle, and the basis 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?
What happened? From one isosceles triangle we get two rectangular ones.
This is already good, but this will happen in any, even the most “oblique” triangle.
How is the picture different for an isosceles triangle? Look again:
Well, firstly, of course, it is not enough for these strange mathematicians to just see - they must certainly prove. Otherwise, suddenly these triangles are slightly different, but we will consider them the same.
But don't worry: in this case, proving is almost as easy as seeing.
Shall we start? Look closely, we have:
And that means! Why? Yes, we will simply find and, and from the Pythagorean theorem (remembering at the same time that)
Are you sure? Well, now we have
And on three sides - the easiest (third) sign of equality of triangles.
Well, our isosceles triangle has divided into two identical rectangular ones.
See how interesting it is? It turned out that:
How do mathematicians usually talk about this? Let's go in order:
(Remember here that the median is a line drawn from a vertex that divides the side in half, and the bisector is the angle.)
Well, here we discussed what good things can be seen if given an isosceles triangle. We deduced that in an isosceles triangle the angles at the base 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 the statements the other way around. This, of course, does not always happen, but an isosceles triangle is still a great thing! What happens after the “turnover”?
Well, look:
If the height and median coincide, then:
If the height and bisector coincide, then: 
If the bisector and the 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 a triangle exactly isosceles and you can draw the height and….(The House That Jack Built…).
- If it turns out that the height is divided in half, then the triangle is isosceles with all the ensuing bonuses.
- If it turns out that the height divides the angle between the floors - it is also isosceles!
- If a bisector divides a side in half or a median divides an angle, then this also happens only in an isosceles triangle
Let's see what it looks like in tasks.
Problem 1(the simplest)
In a triangle, sides and are equal, a. Find.
We decide:
First the drawing.
What is the basis here? Certainly, .
Let's remember what if, then and.
Updated drawing:
Let's denote by. What is the sum of the angles of a triangle? ?
We use:
That's answer: .
Not difficult, right? I didn't even have to adjust the height.
Problem 2(Also not very tricky, but we need to repeat the topic)
In a triangle, . Find.
We decide:
The triangle is isosceles! We draw the height (this is the trick with which everything will be decided now).
Now let’s “cross out from life”, let’s just look at it.
So, we have:
Let's remember the table values of cosines (well, or look at the cheat sheet...)
All that remains is to find: .
Answer: .
Note that we here Very required knowledge regarding right triangles and “tabular” sines and cosines. Very often this happens: the topics , “Isosceles triangle” and in problems go together, but are not very friendly with other topics.
Isosceles triangle. Average level.
These two equal sides are called 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 use one picture to understand why this happens. Let's draw a height from a point.
This means that all corresponding elements are equal.
All! In one fell swoop (height) they proved all the statements at once.
And remember: to solve a problem about an isosceles triangle, it is often very useful to lower the height to the base of the isosceles triangle and divide it into two equal right triangles.
Signs of an isosceles triangle
The converse statements are also true:
Almost all of these statements can again be proven “in one fell swoop.”
1. So, let in turned out to be equal and.
Let's check the height. Then
2. a) Now let in some triangle height and bisector coincide.
2. b) And if the height and median coincide? Everything is almost the same, no more complicated!
 |
- on two sides |
2. c) But if there is no height, which is lowered to the base of an isosceles triangle, then there are no initially right triangles. Badly!
But there is a way out - read it in the next level of the theory, since the proof here is more complicated, but for now just remember that if the median and bisector coincide, then the triangle will also turn out to be isosceles, and the height will still coincide with these bisector and median.
Let's summarize:
- If the triangle is isosceles, then the angles at the base are equal, and the altitude, 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, altitude) 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 a certain triangle two angles are equal, then it is isosceles.
- If in some triangle they 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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The properties of an isosceles triangle are expressed by the following theorems.
Theorem 1. In an isosceles triangle, the angles at the base are equal.
Theorem 2. In an isosceles triangle, the bisector drawn to the base is the median and altitude.
Theorem 3. In an isosceles triangle, the median drawn to the base is the bisector and the altitude.
Theorem 4. In an isosceles triangle, the altitude drawn to the base is the bisector and the median.
Let us prove one of them, for example Theorem 2.5.
Proof. Let us 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 according to the first sign of equality of triangles (AB = AC by condition, AD is a common side, ∠ 1 = ∠ 2, since AD is a bisector). From the equality of these triangles it follows that ∠ B = ∠ C. The theorem is proven.
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 congruent (Fig. 2).
Comment. The sentences established in examples 1 and 2 express the properties of the perpendicular bisector of a segment. From these proposals it follows that perpendicular bisectors to the sides of a triangle intersect at one point.
Example 1. Prove that a point in the plane equidistant from the ends of a segment lies on the perpendicular bisector to this segment.
Solution. Let point M be equidistant from the ends of segment AB (Fig. 3), i.e. AM = BM.
Then Δ AMV is isosceles. Let us draw a straight line p through the point M and the midpoint O of the segment AB. By construction, the segment MO is the median of the isosceles triangle AMB, and therefore (Theorem 3), and the height, i.e., the straight line MO, is the perpendicular bisector to the segment AB.
Example 2. Prove that each point of the perpendicular bisector to a segment is equidistant from its ends.
Solution. Let p be the perpendicular bisector to segment AB and point O be the midpoint of segment AB (see Fig. 3).
Consider an arbitrary point M lying on the straight line p. Let's draw segments AM and BM. Triangles AOM and BOM are equal, since their angles at vertex O are right, leg OM is common, and leg OA is equal to leg OB by condition. From the equality of triangles AOM and BOM it follows that AM = BM.
Example 3. In triangle ABC (see Fig. 4) AB = 10 cm, BC = 9 cm, AC = 7 cm; in triangle DEF DE = 7 cm, EF = 10 cm, FD = 9 cm.
Compare triangles ABC and DEF. Find the corresponding equal angles.
Solution. These triangles are equal according to the third criterion. Correspondingly, equal angles: A and E (lie opposite equal sides BC and FD), B and F (lie opposite equal sides AC and DE), C and D (lie opposite 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 according to the third criterion (AB = DC, BC = AD by condition and side AC is common). From the equality of these triangles it follows that ∠ B = ∠ D, but angle B is equal to 100°, which means angle D is equal to 100°.
Example 5. In an isosceles triangle ABC with base AC, the exterior angle at vertex C is 123°. Find the size of angle ABC. Give your answer in degrees.
Video solution.
