In an equilibried triangle, the parties are equal. Isosceles triangle

29.09.2019

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Isosceles triangle - This is a triangle, in which two sides are equal to each other in length. The equal parties are called side, and the latter is the basis. By definition, the correct triangle is also an equally chagrined, but the opposite statement is incorrect.

Properties

Corners, opposite to equal sides of an equilibried triangle, are equal to each other. Also equal to bisector, medians and heights conducted from these corners.
Bisectrix, median, height and a middle perpendicular, conducted to the base, coincide with each other. Centers inscribed and described circles lie on this line.
Corners, opposite to equal parties, are always sharp (follows from their equality).

Let be a. - the length of two equal sides of an equifiable triangle, b. - the length of the third party, α and β - appropriate angles, R. - radius of the described circle, r. - The radius inscribed.

The parties can be found as follows:

Corners can be expressed in the following ways:

The perimeter of an equilibrium triangle can be calculated by any of the following ways:

The area of \u200b\u200bthe triangle can be calculated in one of the following ways:

(Formula of Geron).

Signs

Two triangle angle are equal.
The height coincides with the median.
The height coincides with the bisector.
Bissectrix coincides with the median.
Two heights are equal.
Two medians are equal.
Two bisector are equal (Steiner Theorem - Lemus).

It is given: Δ ABC is a befled with the base of the AU, VK bisectris, AC \u003d 46cm. Find: AK. (Fig.4)
It is given: Δ ABC is a befled with the base of the AU, VK height, ABC \u003d 46 0. Find: AVK. (Fig. 5)
It is given: Δ with BD isceived with the base B C, DA median, VS \u003d 120 0. Find: adb. (Fig. 6)

7th grade

Properties of an equally chained triangle

Three ways lead to knowledge:

Reflectance path is the most noble path

Imitation path is the easiest way,

And the path of experience is the path is the most bitter.

Confucius.

In an equilibried triangle, the angles at the base are equal.

Dano: ABC is a preceded

Prove

Evidence:

1. We carry out bisectris BD angle V.

2. Consider Δ AB D and Δ CBD:

AB \u003d BC (under the condition),

In D - general side,

∠ A BD \u003d ∠ with BD

Δ AVD \u003d ΔCBD (1 sign of equality of triangles)

3. In equal triangles, corresponding angles are equal to ∠ A \u003d ∠ S.

In an equilibried triangle of bisector, conducted to the base, is median and height.

Given: ABC is a preceded,

BUT D - bisector .

Prove BUT D. - Height,

BUT D. - Median.

Evidence:

1) Consider:

Δ Bad \u003d ΔCad (1 sign of the equality of triangles).

2) in equal triangles, respectively, the respective side and corners are equal

1 \u003d 2 \u003d 90 ° (adjacent angles).

Therefore, AD is median and height Δ ABC.

Solving tasks.

Savrasova S.M., Yarstreyskysky G.A. "Exercises on planimetry on ready-made drawings"

110

Solving tasks.

Danched: Av \u003d in C, 1 \u003d 130 0.

L. S. Atanasyan. Geometry 7-9 No. 112.

Solving tasks.

Find: AB D.

Triangle

ABC - Equal

In D - Median

So, in D - bisector

40 0

CM. Savrasova, G.A. Yatrevilky "Exercises on the finished drawings"

Homework:

p. 19 (p. 35 - 36), No. 109, 112, 118.

At this lesson, the theme "Equal triangle and its properties" will be considered. You will learn what looks like and equilateral and equilateral triangles. Prove the theorem on the equality of the angles at the base of an equifiable triangle. Consider also the theorem on the bisector (median and height) conducted to the founding of an inaccessible triangle. At the end of the lesson, you will analyze two tasks using the definition and properties of an equally triangle.

Definition:Equalued A triangle is called, in which two sides are equal.

Fig. 1. Equal triangle

AB \u003d AC - side sides. Sun is the basis.

The area of \u200b\u200ban equilibried triangle is equal to half of the product of its base to height.

Definition:Equilateral A triangle is called, in which all three sides are equal.

Fig. 2. Equipical triangle

AB \u003d Sun \u003d sa.

Theorem 1: In an equilibried triangle, the angles at the base are equal.

Given: AU \u003d AU.

Prove ∠V \u003d ∠С.

Fig. 3. Drawing to theorem

Evidence: ABC triangle \u003d DR triangle on the first sign (on two equal parties and the corner between them). From the equality of triangles, equality of all relevant elements follows. So, ∠v \u003d ∠c, which was required to prove.

Theorem 2: In an equally traded triangle bisectorconducted to the ground is median and height.

Given: AU \u003d AC, ∠1 \u003d ∠2.

Prove Cd \u003d dc, ad perpendicular to BC.

Fig. 4. Drawing to Theorem 2

Evidence: ADB triangle \u003d ADC triangle on the first basis (AD - total, AV \u003d AC for condition, ∠Bad \u003d ∠DAC). From the equality of triangles, equality of all relevant elements follows. BD \u003d DC, as they lie against equal corners. So, AD is median. Also ∠3 \u003d ∠4, as they lie against the equal parties. But, moreover, they are equal in the amount. Consequently, ∠3 \u003d ∠4 \u003d. So, AD is the height of the triangle, which was required to prove.

In the only case a \u003d b \u003d. In this case, direct AC and CD are called perpendicular.

Since bisector, height and median is the same segment, the following statements are both:

The height of an inaccessible triangle, conducted to the base, is median and bisector.

The median is a preceded triangle, conducted to the base, is height and bisector.

Example 1: In an equilibried triangle, the base is two times less than the side, and the perimeter is 50 cm. Find the sides of the triangle.

Given: AU \u003d AC, Sun \u003d AC. P \u003d 50 cm.

To find: Sun, AS, AV.

Decision:

Fig. 5. Drawing for example 1

Denote the base of the aircraft as a, then av \u003d ac \u003d 2a.

2a + 2a + a \u003d 50.

5a \u003d 50, a \u003d 10.

Answer: Sun \u003d 10 cm, AC \u003d AB \u003d 20 cm.

Example 2: Prove that in the equilateral triangle all corners are equal.

Given: AB \u003d Sun \u003d sa.

Prove ∠A \u003d ∠B \u003d ∠С.

Evidence:

Fig. 6. Drawing for example

∠B \u003d ∠C, since Av \u003d ac, and ∠a \u003d ∠, since the speaker \u003d sun.

Consequently, ∠a \u003d ∠v \u003d ∠c, which was required to prove.

Answer: Proved.

In today's lesson, we looked at an equifiable triangle, studied its basic properties. In the next lesson, we cut the challenges on the topic of an inaccessible triangle, to calculate the area of \u200b\u200ban inaccessible and equilateral triangle.

Alexandrov A.D., Werner A.L., Ryzhik V.I. and others. Geometry 7. - M.: Enlightenment.
Atanasyan L.S., Butuzov V.F., Kadomtsev S.B. et al. Geometry 7. 5th ed. - M.: Enlightenment.
Butuzov V.F., Kadomtsev S.B., Prasolova V.V. Geometry 7 / V.F. Butuzov, S.B. Kadomtsev, V.V. Prasolova, ed. Sadovnichny V.A. - M.: Enlightenment, 2010.

Dictionaries and encyclopedias at the Academician ().
Festival of pedagogical idea "Open lesson" ().
Kaknauchit.ru ().

1. No. 29. Butuzov V.F., Kadomtsev S.B., Prasolova V.V. Geometry 7 / V.F. Butuzov, S.B. Kadomtsev, V.V. Prasolova, ed. Sadovnichny V.A. - M.: Enlightenment, 2010.

2. The perimeter of an equilibrium triangle is 35 cm, and the base is three times less lateral side. Find the sides of the triangle.

3. Danarily: AV \u003d Sun. Prove that ∠1 \u003d ∠2.

4. The perimeter of an equilibried triangle is 20 cm, one of its sides is twice the other. Find the sides of the triangle. How many solutions have a task?

The first historians of our civilization are the ancient Greeks - refer to Egypt as a place of the origin of geometry. It is difficult to disagree with them, knowing, with which amazing accuracy is the giant tomb of the pharaohs. The mutual layout of the planes of the pyramids, their proportions, orientation on the sides of the world - to achieve such perfection would be unthinkable, not knowing the basics of geometry.

The word "geometry" itself can be translated as "earth measurement". And the word "earth" acts not as a planet - part of the solar system, but as a plane. The markup of the area under the maintenance of agriculture is likely to be the most initial basis of science on geometric figures, their types and properties.

The triangle is the simplest spatial figure of planimetry, containing only three points - vertices (no less). The basis of the foundation may be because it is hung in it something mysterious and ancient. The Oco Oco inside the triangle is one of the earliest of the famous occult signs, and the geography of its distribution and the time frame simply amazing imagination. From the ancient Egyptian, Sumerian, Aztec and other civilizations to more modern communities of amateurs of occultism scattered around the globe.

What are the triangles

An ordinary versatile triangle is a closed geometric figure, consisting of three segments of different lengths and three corners, none of which is not direct. In addition to him, there are several special species.

The triangle acutely has all the angles of less than 90 degrees. In other words, all the angles of such a triangle are sharp.

The rectangular triangle over which schoolchildren were crying due to theorem's abundance, has one angle with a value of 90 degrees or, as it is also called direct.

The stupid triangle is characterized by the fact that one of his corners is stupid, that is, its value is more than 90 degrees.

The equilateral triangle has three sides of the same length. Such a figure is also equal to all angles.

Finally, at an equifiable triangle of three sides, two are among themselves.

Distinctive features

The properties of an equifiable triangle determine its main, most importantly, the difference is the equality of both sides. These parties are considered to be called the hips (or, more often, side sides), but the third party is called the "base".

On the considered figure a \u003d b.

The second sign of an equifiable triangle follows from the sinus theorem. Since the sides of A and B are equal to the sines of their opposite angles:

a / SIN γ \u003d b / sin α, from where we have: sin γ \u003d sin α.

From the equality of sinuses, equality of the angles is followed: γ \u003d α.

So, the second sign of an equilibrium triangle is the equality of two angles adjacent to the base.

Third sign. The triangle distinguish such elements as height, bisector and median.

If in the process of solving the problem it turns out that in the triangle under consideration, two any of these elements coincide: height with bisector; Bissectrix with median; Median with a height - unambiguously we can conclude that the triangle is wasosbered.

Geometric properties of Figure

1. Properties of an isced triangle. One of the distinctive features of the figure is the equality of angles adjacent to the base:

<ВАС = <ВСА.

2. Another property is discussed above: the median, bisector and height in an equilibried triangle coincide if they are built from its vertices to the base.

3. Equality of bissectris conducted from vertices at the base:

If ae is the bisector of the angle of you, and the CD is bisectrice of the BCA angle, then: ae \u003d dc.

4. The properties of an equifiable triangle also provide equality of heights that are carried out from the vertices at the base.

If you construct the height of the ABS triangle (where AV \u003d Sun) from the vertices A and C, then the obtained CD segments and ae will be equal.

5. The medians spent from corners at the base will also be equal.

So, if ae and dc are medians, that is, ad \u003d db, and be \u003d ec, then Ae \u003d DC.

The height of an inaccessible triangle

Equality of side sides and corners with them introduces some features in calculating the lengths of the elements of the figure under consideration.

The height in an equilibried triangle divides the figure on 2 symmetric rectangular triangles, with hypotenuses in which side sides are. The height in this case is determined according to the Pythagora theorem like catat.

The triangle can be equal to all three sides, then it will be called equilateral. The height in the 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 and other way, for example, knowing the base and the angle adjacent to it.

Median is a preceded triangle

The type of triangle is considered, due to the geometric characteristics, is solved quite simply at the minimum set of source data. Since the median in an equilibried triangle is equal to its height, and its bisector, the algorithm of its definition does not differ from the order of calculating these elements.

For example, it is possible to determine the length of the median in the well-known side and the magnitude of the angle at the top.

How to determine perimeter

Since the planimetric figure in question, the two sides are always equal, then it is enough to know the length of the base and the length of one of the parties to determine the perimeter.

Consider an example when you need to determine the perimeter of the triangle on the well-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 Pythagora theorem as a rectangular hypotenus. Its length is equal to the root square of the sum of the square of the height and square of half the base.

Square of an equally chained triangle

It does not cause, as a rule, difficulties and calculation of an equally-free triangle area. The universal rule of determining the area of \u200b\u200bthe triangle as half of the product of the base on its height is applicable, of course, in our case. However, the properties of an equilibried triangle again facilitate the task.

Suppose that the height and angle adjacent to the base are known. It is necessary to determine the area of \u200b\u200bthe figure. You can do this in this way.

Since the sum of the angles of any triangle is 180 °, then it is not difficult to determine the corner. Next, using the proportion compiled according to the sinus theorem, the length of the triangle base is determined. Everything, base and height - sufficient data to determine the area - are available.

Other properties of an equilibrium triangle

The position of the center of the circle described around an equilibried triangle depends on the magnitude of the angle of the vertex. So, if an anoscele triangle is acute, the center of the circle is located inside the figure.

The center of the circle, which is described around a stupid iscessed triangle, lies outside it. And, finally, if the magnitude of the angle at the top is 90 °, the center lies exactly in the middle of the base, and through the base itself passes the diameter of the circle.

In order to determine the radius of the circle described near an equilibried triangle, it suffices to divide the lateral side to the double cosine of half the angle of the corner at the vertex.

In an equilibried triangle, the parties are equal. Isosceles triangle

Read also

Properties

Signs

see also

Watch what is an "equally chaired triangle" in other dictionaries:

What are the triangles

Distinctive features

Geometric properties of Figure

The height of an inaccessible triangle

Median is a preceded triangle

How to determine perimeter

Square of an equally chained triangle

Other properties of an equilibrium triangle