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 very often turn out to be the main actors in the tasks of the Unified State Examination of 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, and now we will consider 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, look at 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, two right-angled ones turned out.
This is already good, but this will happen in any, the most “oblique” triangle.
What is the difference between the picture for an isosceles triangle? Look again:
Well, firstly, 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.
Shall we start? Look carefully, we have:
And, therefore,! Why? Yes, we just 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 the equality of triangles.
Well, our isosceles triangle is divided into two identical rectangular ones.
See how interesting? It turned out that:
How is it customary for mathematicians to talk about this? Let's go in order:
(We recall here that the median is a line drawn from the vertex that bisects the side, and the bisector is the angle.)
Well, here we discussed what good can be seen if given an isosceles triangle. We have deduced that in an isosceles triangle the angles at the base are equal, and the height, bisector and median drawn to the base are the same.
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 on the contrary. This, of course, does not always happen, but an isosceles triangle is still a great thing! What happens after the "reversal"?
Well look here:
If height and median are the same, then:
If the height and bisector are the same, then: 
If the bisector and median are the same, then:
Well, do not forget and use:
- If an isosceles triangle is given, feel free to draw a height, get two right triangles and solve the problem already about a right triangle.
- If given that two angles are equal, then the triangle exactly isosceles and you can draw a height and .... (The house that Jack built ...).
- If it turned out that the height is divided in half by the side, then the triangle is isosceles with all the ensuing bonuses.
- If it turned out that the height divided the angle to the floors - also isosceles!
- If the bisector divided the side in half or the median - the angle, then this also happens only in an isosceles triangle
Let's see how it looks in tasks.
Task 1(the simplest)
In a triangle, the sides and are equal, a. To find.
We decide:
First a drawing.
What is the basis here? Certainly, .
We recall that if, then and.
Updated drawing:
Let's designate for. What is the sum of the angles of the triangle? ?
We use:
That's answer: .
Easy, right? I didn't even have to go high.
Task 2(Also not very tricky, but you need to repeat the theme)
In a triangle, To find.
We decide:
The triangle is isosceles! We draw the height (this is the focus, with the help of which everything will be decided now).
Now "we delete from life", we will consider only.
So, in we have:
We recall the tabular values of cosines (well, or look at the cheat sheet ...)
It remains to find: .
Answer: .
Note that we are here very required knowledge regarding the right triangle and the "tabular" sines and cosines. Very often this happens: the topics, “Isosceles Triangle” and in puzzles go in bundles, but they are not very friendly with other topics.
Isosceles triangle. Middle level.
These two equal sides called sides, a the third side is the base of an isosceles triangle.
Look at the picture: and - the sides, - the base of an isosceles triangle.
Let's see in one picture why this is so. Draw a height from a point.
This means that all corresponding elements are equal.
Everything! In one fell swoop (height) all the statements were proved at once.
And you 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 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 v turned out to be equal and.
Let's take the height. Then
2. a) Now let in some triangle same height and bisector.
2. b) And if the height and median are the same? Everything is almost the same, nothing more complicated!
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- 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 triangles. Badly!
But there is a way out - read it in the next level of theory, because the proof is more complicated here, but for now just remember that if the median and the bisector coincide, then the triangle will also be isosceles, and the height will still coincide with these bisector and median.
To summarize:
- If the triangle is isosceles, then the angles at the base are equal, and the height, bisector and median drawn to the base are the same.
- 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 a triangle has two equal angles, then it is isosceles.
- If in some triangle coincide:
a) height and bisector or
b) height and median or
in) median and bisector,
drawn to one side, then such a triangle is isosceles.
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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 accuracy the giant tombs of the pharaohs were erected. The mutual arrangement of the planes of the pyramids, their proportions, orientation to the cardinal points - it would be unthinkable to achieve such perfection without knowing the basics of geometry.
The very word "geometry" can be translated as "measurement of the earth." Moreover, the word "earth" appears not as a planet - part of the solar system, but as a plane. The marking of areas for agriculture, most likely, is the very original basis of the science of geometric shapes, their types and properties.
A triangle is the simplest spatial figure of planimetry, containing only three points - vertices (there is no less). The foundation of foundations, perhaps, is why something mysterious and ancient seems to be in it. 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 around 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 is straight. In addition to it, there are several special types.
An acute triangle has all angles less than 90 degrees. In other words, all angles of such a triangle are acute.
A right-angled triangle, over which schoolchildren have cried at all times because of the abundance of theorems, has one angle with a value of 90 degrees, or, as it is also called, a right one.
An obtuse triangle is distinguished by the fact that one of its angles is obtuse, that is, its value is more than 90 degrees.
An equilateral triangle has three sides of the same length. In such a figure, all angles are also equal.
And finally, in an isosceles triangle of 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 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 sign of an isosceles triangle follows from the sine theorem. Since the 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.
From the equality of the sines follows the equality of the 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, elements such as height, bisector and median are distinguished.
If in the process of solving the problem it turns out that in the triangle under consideration, 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. Another property discussed above: the median, bisector and height in an isosceles triangle are the same if they are built from its top to the base.
3. The equality of the 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 build the heights of the triangle ABC (where AB = BC) from the 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 turn out to 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 at them introduces some features 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 hypotenuses of which are 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, knowing the base and the angle adjacent to it.
Median of an isosceles triangle
The type of triangle under consideration, due to 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 determining it 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 angle at the vertex.
How to determine the perimeter
Since the planimetric figure under consideration has two sides always equal, to determine the perimeter it is enough to know the length of the base and the length of one of the sides.
Consider an example when you need to determine the perimeter of a triangle given the 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 determined 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
Does not cause, as a rule, difficulties and the calculation of 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 is applicable, of course, in our case. However, the properties of an isosceles triangle again make the task easier.
Let us assume that we know the height and the angle adjacent to the base. You need 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 magnitude of the angle. Further, using the proportion drawn up according to the sine theorem, 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 angle of the vertex. So, if an isosceles triangle is acute-angled, 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 lateral side by twice the cosine of half the angle at the vertex.
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 drawn to the base is the median and height.
Theorem 3. In an isosceles triangle, the median drawn to the base is the bisector and 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 according to the first sign of equality of triangles (AB = AC by condition, AD is the common side, ∠ 1 = ∠ 2, since AD is the bisector). It follows from the equality of these triangles that ∠ B = ∠ C. The theorem is proved.
Using Theorem 1, we establish the following theorem.
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 established in examples 1 and 2 express the properties of the perpendicular bisector to the segment. It follows from these proposals that the perpendicular bisectors of the sides of a 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 bisector to this segment.
Decision. Let the point M be equidistant from the ends of the segment AB (Fig. 3), i.e. AM = VM.
Then ΔAMV is isosceles. Let us draw a 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 of a segment is equidistant from its ends.
Decision. 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 line p. Let's draw segments AM and VM. Triangles AOM and VOM are equal, since their angles at the vertex O are straight, the leg OM is common, and the leg OA is equal to the leg OB by condition. From the equality of triangles AOM and BOM it follows that AM = BM.
Example 3 In the triangle ABC (see Fig. 4) AB \u003d 10 cm, BC \u003d 9 cm, AC \u003d 7 cm; in triangle DEF DE = 7 cm, EF = 10 cm, FD = 9 cm.
Compare triangles ABC and DEF. Find correspondingly equal angles.
Decision. These triangles are equal in the third criterion. Accordingly, equal angles: A and E (they lie opposite the equal sides BC and FD), B and F (they lie opposite the equal sides AC and DE), C and D (they lie opposite the equal sides AB and EF).
Example 4 In figure 5 AB = DC, BC = AD, ∠B = 100°.
Find angle D.
Decision. Consider triangles ABC and ADC. They are equal in the third feature (AB = DC, BC = AD by condition and side AC is common). From the equality of these triangles it follows that ∠ B = ∠ D, but the angle B is 100°, hence the angle D is 100°.
Example 5 In an isosceles triangle ABC with base AC, the exterior angle at vertex C is 123°. Find the angle ABC. Give your answer in degrees.
Video solution.
