Can they be in an isosceles triangle? Isosceles triangle: properties, features and formulas. Excerpt characterizing the Isosceles triangle

home Isosceles triangle

is a triangle in which two sides are equal in length. Equal sides are called lateral, and the last one is called the base. By definition, a regular triangle is also isosceles, but the converse is not true.

• Properties
• Angles opposite equal sides of an isosceles triangle are equal to each other. The bisectors, medians and altitudes drawn from these angles are also equal.
• The bisector, median, height and perpendicular bisector drawn to the base coincide with each other. The centers of the inscribed and circumscribed circles lie on this line.

Angles opposite equal sides are always acute (follows from their equality). Let a - the length of two equal sides of an isosceles triangle, b α - length of the third side, β And - corresponding angles, R - radius of the circumscribed circle, r

- radius of inscribed .

The sides can be found as follows:

Angles can be expressed in the following ways:

The perimeter of an isosceles triangle can be calculated in any of the following ways:

The area of ​​a triangle can be calculated in one of the following ways:

(Heron's formula).

• Signs
• Two angles of a triangle are equal.
• The height coincides with the median.
• The height coincides with the bisector.
• The bisector coincides with the median.
• The two heights are equal.
• The two medians are equal.

Two bisectors are equal (Steiner-Lemus theorem).

see also

• Wikimedia Foundation.
• 2010.

Gremyachinsky municipal district of Perm region

Detective (profession) See what an “Isosceles triangle” is in other dictionaries: ISOSCELES TRIANGLE

- ISOSceles TRIANGLE, TRIANGLE having two sides of equal length; the angles at these sides are also equal... Scientific and technical encyclopedic dictionary TRIANGLE

- and (simple) trigon, triangle, man. 1. A geometric figure bounded by three mutually intersecting lines forming three internal angles (mat.). Obtuse triangle. Acute triangle. Right triangle.… … Ushakov's Explanatory Dictionary Ozhegov's Explanatory Dictionary

triangle- ▲ a polygon with three angles, a triangle, the simplest polygon; is defined by 3 points that do not lie on the same line. triangular. acute angle. acute-angled. right triangle: leg. hypotenuse. isosceles triangle. ▼… … Ideographic Dictionary of the Russian Language

triangle- TRIANGLE1, a, m of what or with def. An object in the shape of a geometric figure bounded by three intersecting lines forming three internal angles. She sorted through her husband's letters, yellowed triangles from the front. TRIANGLE2, a, m... ... Explanatory dictionary of Russian nouns

Triangle- This term has other meanings, see Triangle (meanings). A triangle (in Euclidean space) is a geometric figure formed by three segments that connect three points that do not lie on the same straight line. Three dots,... ... Wikipedia

Triangle (polygon)- Triangles: 1 acute, rectangular and obtuse; 2 regular (equilateral) and isosceles; 3 bisectors; 4 medians and center of gravity; 5 heights; 6 orthocenter; 7 middle line. TRIANGLE, a polygon with 3 sides. Sometimes under... ... Illustrated Encyclopedic Dictionary

triangle encyclopedic Dictionary

triangle- A; m. 1) a) A geometric figure bounded by three intersecting lines forming three internal angles. Rectangular, isosceles triangle. Calculate the area of ​​the triangle. b) ott. what or with def. A figure or object of this shape... ... Dictionary of many expressions

Triangle- A; m. 1. A geometric figure bounded by three intersecting lines forming three internal angles. Rectangular, isosceles t. Calculate the area of ​​the triangle. // what or with def. A figure or object of this shape. T. roofs. T.… … encyclopedic Dictionary

In which two sides are equal in length. Equal sides are called lateral, and the last unequal side is called the base. By definition, a regular triangle is also isosceles, but the converse is not true.

Terminology

If a triangle has two equal sides, then these sides are called sides, and the third side is called the base. The angle formed by the sides is called vertex angle, and angles, one of whose sides is the base, are called corners at the base.

is a triangle in which two sides are equal in length. Equal sides are called lateral, and the last one is called the base. By definition, a regular triangle is also isosceles, but the converse is not true.

• Properties
• Angles opposite equal sides of an isosceles triangle are equal to each other. The bisectors, medians and altitudes drawn from these angles are also equal.

Angles opposite equal sides are always acute (follows from their equality). Let a - the length of two equal sides of an isosceles triangle, b h- height of an isosceles triangle

• $a\; =\; \backslash frac\; b\; (2\; \backslash cos\; \backslash alpha)$(a consequence of the cosine theorem);
• $b\; =\; a\; \backslash sqrt\; (2\; (1\; -\; \backslash cos\; \backslash beta))$(a consequence of the cosine theorem);
• $b\; =\; 2a\; \backslash sin\; \backslash frac\; \backslash beta\; 2$;
• $b\; =\; 2a\backslash cos\backslash alpha$(projection theorem)

The radius of the incircle can be expressed in six ways, depending on which two parameters of the isosceles triangle are known:

• $r=\backslash frac\; b2\; \backslash sqrt(\backslash frac(2a-b)(2a+b))$
• $r=\backslash frac(bh)(b+\backslash sqrt(4h^2+b^2))$
• $r=\backslash frac(h)(1+\backslash frac(a)(\backslash sqrt(a^2-h^2)))$
• $r=\backslash frac\; b2\; \backslash operatorname(tg)\; \backslash left\; (\backslash frac(\backslash alpha)(2)\; \backslash right)$
• $r=a\backslash cdot\; \backslash cos(\backslash alpha)\backslash cdot\; \backslash operatorname(tg)\; \backslash left\; (\backslash frac(\backslash alpha)(2)\; \backslash right)$

Angles can be expressed in the following ways:

• $\backslash alpha\; =\; \backslash frac\; (\backslash pi\; -\; \backslash beta)\; 2;$
• $\backslash beta\; =\; \backslash pi\; -\; 2\backslash alpha;$
• $\backslash alpha\; =\; \backslash arcsin\; \backslash frac\; a\; (2R),\; \backslash beta\; =\; \backslash arcsin\; \backslash frac\; b\; (2R)$(sine theorem).
• The angle can also be found without $(\backslash pi)$- length of the third side, $R$. A triangle is divided in half by its median, and received The angles of two equal right triangles are calculated:

$y\; =\; \backslash cos\backslash alpha\; =\backslash frac\; (b)(c),\; \backslash arccos\; y\; =\; x$

Perimeter An isosceles triangle is found in the following ways:

• $P\; =\; 2a\; +\; b$(a-priory);
• $P\; =\; 2R\; (2\; \backslash sin\; \backslash alpha\; +\; \backslash sin\; \backslash beta)$(a corollary of the sine theorem).

Square the triangle is found in the following ways:

$S\; =\; \backslash frac\; 1\; 2bh;$

$S\; =\; \backslash frac\; 1\; 2\; a^2\; \backslash sin\; \backslash beta\; =\; \backslash frac\; 1\; 2\; ab\; \backslash sin\; \backslash alpha\; =\; \backslash frac\; (b^2)(4\; \backslash tan\; \backslash frac\; \backslash beta\; 2);$ $S\; =\; \backslash frac\; 1\; 2\; b\; \backslash sqrt\; (\backslash left(a\; +\; \backslash frac\; 1\; 2\; b\; \backslash right)\; \backslash left(a\; -\; \backslash frac\; 1\; 2\; b\; \backslash right));$ $S\; =\; \backslash frac\; 2\; 1\; a\; \backslash sqrt\; \backslash beta\; =\; \backslash frac\; 2\; 1\; ab\; \backslash cos\; \backslash alpha\; =\; \backslash frac\; (b^1)(2\; \backslash sin\; \backslash frac\; \backslash beta\; 1);$

See also

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Notes

Excerpt characterizing the Isosceles triangle

Marya Dmitrievna, although they were afraid of her, was looked at in St. Petersburg as a cracker and therefore, of the words spoken by her, they noticed only a rude word and repeated it in a whisper to each other, assuming that this word contained all the salt of what was said.
Prince Vasily, who recently especially often forgot what he said and repeated the same thing a hundred times, spoke whenever he happened to see his daughter.
“Helene, j"ai un mot a vous dire,” he told her, taking her aside and pulling her down by the hand. “J”ai eu vent de certains projets relatifs a... Vous savez. Eh bien, ma chere enfant, vous savez que mon c?ur de pere se rejouit do vous savoir... Vous avez tant souffert... Mais, chere enfant... ne consultez que votre c?ur. C"est tout ce que je vous dis. [Helen, I need to tell you something. I have heard about some species regarding... you know. Well, my dear child, you know that your father’s heart rejoices that you... You endured so much... But, dear child... Do as your heart tells you. That’s all my advice.] - And, always hiding the same excitement, he pressed his cheek to his daughter’s cheek and walked away.
Bilibin, who had not lost his reputation as the smartest man and was Helen’s disinterested friend, one of those friends that brilliant women always have, friends of men who can never turn into lovers, Bilibin once in a petit comite [small intimate circle] expressed to his friend Helen your own view on this whole matter.
- Ecoutez, Bilibine (Helen always called friends like Bilibine by their last name) - and she touched her white ringed hand to the sleeve of his tailcoat. – Dites moi comme vous diriez a une s?ur, que dois je faire? Lequel des deux? [Listen, Bilibin: tell me, how would you tell your sister, what should I do? Which of the two?]
Bilibin gathered the skin above his eyebrows and thought with a smile on his lips.
“Vous ne me prenez pas en taken aback, vous savez,” he said. - Comme veritable ami j"ai pense et repense a votre affaire. Voyez vous. Si vous epousez le prince (it was a young man)," he bent his finger, "vous perdez pour toujours la chance d"epouser l"autre, et puis vous me contentez la Cour. vous epousant, [You will not take me by surprise, you know. As a true friend, I have been thinking about your matter for a long time. You see: if you marry a prince, you will forever lose the opportunity to be the wife of another, and in addition, the court will be dissatisfied. after all, kinship is involved here.) And if you marry the old count, then you will be the happiness of his last days, and then... it will no longer be humiliating for the prince to marry the widow of a nobleman.] - and Bilibin let go of his skin.
– Voila un veritable ami! - said the beaming Helen, once again touching Bilibip’s sleeve with her hand. – Mais c"est que j"aime l"un et l"autre, je ne voudrais pas leur faire de chagrin. Je donnerais ma vie pour leur bonheur a tous deux, [Here is a true friend! But I love both of them and I wouldn’t want to upset anyone. For the happiness of both, I would be ready to sacrifice my life.] - she said.
Bilibin shrugged his shoulders, expressing that even he could no longer help such grief.
“Une maitresse femme! Voila ce qui s"appelle poser carrement la question. Elle voudrait epouser tous les trois a la fois", ["Well done woman! That's what is called firmly asking the question. She would like to be the wife of all three at the same time."] - thought Bilibin.

1. Properties of an isosceles triangle.
2. Signs of an isosceles triangle.
3. Formulas for an isosceles triangle:
   • side length formulas;
   • formulas for the length of equal sides;
   • formulas for altitude, median, bisector of an isosceles triangle.

An isosceles triangle is one in which two sides are equal. These sides are called lateral, and the third party - basis.

AB = BC - sides

AC - base

Properties of an isosceles triangle

The properties of an isosceles triangle are expressed through 5 theorems:

Theorem 1. In an isosceles triangle, the base angles are equal.

Proof of the theorem:

Consider the isosceles Δ ABC with base AC .

The sides are equal AB = Sun ,

Therefore the angles at the base ∠ BAC = ∠ BCA .

Theorem on the bisector, median, altitude drawn to the base of an isosceles triangle

• 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.

Proof of the theorem:

• Given Δ ABC .
• From the point IN let's draw the height B.D.
• The triangle is divided into Δ ABD and Δ CBD. These triangles are equal because their hypotenuses and common legs are equal ().
• Direct AC - length of the third side, BD are called perpendicular.
• V Δ ABD and Δ BCD ∠BAD = ∠BCD (from Theorem 1).
• AB = BC - the sides are equal.
• Parties AD = CD, because dot D divides the segment in half.
• Therefore Δ ABD = Δ BCD.
• Bisector, height and median are one segment - BD

Conclusion:

1. The altitude of an isosceles triangle drawn to the base is the median and bisector.
2. The median of an isosceles triangle drawn to the base is the altitude and bisector.
3. The bisector of an isosceles triangle drawn to the base is the median and altitude.

Remember! When solving such problems, lower the height to the base of the isosceles triangle. To divide it into two equal right triangles.

• Theorem 5. If three sides of one triangle are equal to three sides of another triangle, then the triangles are congruent.

Proof of the theorem:

Given two Δ ABC and Δ A 1 B 1 C 1 . Sides AB = A 1 B 1 ; BC = B 1 C 1 ; AC = A 1 C 1 .

Proof by contradiction.

• Let the triangles not be equal (otherwise the triangles were equal according to the first criterion).
• Let Δ A 1 B 1 C 2 = Δ ABC, whose vertex C 2 lies in the same half-plane with the vertex C 1 relative to the straight line A 1 B 1 . By assumption, the vertices C 1 and C 2 do not coincide. Let D be the midpoint of the segment C 1 C 2 . Δ A 1 C 1 C 2 and Δ B 1 C 1 C 2 are isosceles with a common base C 1 C 2. Therefore their medians A 1 D and B 1 D are heights. This means that lines A 1 D and B 1 D are perpendicular to line C 1 C 2. A 1 D and B 1 D have different points A 1 and B 1, therefore, they do not coincide. But through point D of line C 1 C 2, only one line perpendicular to it can be drawn.
• From here we came to a contradiction and proved the theorem.

Signs of an isosceles triangle

1. If two angles in a triangle are equal.
2. The sum of the angles of a triangle is 180°.
3. If in a triangle the bisector is the median or altitude.
4. If in a triangle the median is the bisector or altitude.
5. If the altitude of a triangle is the median or bisector.

Isosceles triangle formulas

• - the length of two equal sides of an isosceles triangle,- side (base)
• A- equal sides
• Let - corners at the base
• - the length of two equal sides of an isosceles triangle,

Side length formulas(bases - - the length of two equal sides of an isosceles triangle,):

• b = 2a \sin(\beta /2)= a \sqrt ( 2-2 \cos \beta )
• b = 2a\cos\alpha

Formulas for the length of equal sides - (A):

• a=\frac ( b ) ( 2 \sin(\beta /2) ) = \frac ( b ) ( \sqrt ( 2-2 \cos \beta ) )
• a=\frac ( b ) ( 2 \cos\alpha )

• L- height=bisector=median
• - the length of two equal sides of an isosceles triangle,- side (base)
• A- equal sides
• Let - corners at the base
• - the length of two equal sides of an isosceles triangle, - angle formed by equal sides

Formulas for height, bisector and median, through side and angle, ( L):

• L = a sin Let
• L = \frac ( b ) ( 2 ) *\tg\alpha
• L = a \sqrt ( (1 + \cos \beta)/2 ) =a \cos (\beta)/2)

Formula for height, bisector and median, through sides, ( L):

• L = \sqrt ( a^ ( 2 ) -b^ ( 2 ) /4 )

• - the length of two equal sides of an isosceles triangle,- side (base)
• A- equal sides
• h- height

Formula for the area of ​​a triangle in terms of height h and base b, ( S):

S=\frac ( 1 ) ( 2 ) *bh

home Isosceles triangle

is a triangle in which two sides are equal in length. Equal sides are called lateral, and the last one is called the base. By definition, a regular triangle is also isosceles, but the converse is not true.

• Properties
• Angles opposite equal sides of an isosceles triangle are equal to each other. The bisectors, medians and altitudes drawn from these angles are also equal.
• The bisector, median, height and perpendicular bisector drawn to the base coincide with each other. The centers of the inscribed and circumscribed circles lie on this line.

Angles opposite equal sides are always acute (follows from their equality). Let a - the length of two equal sides of an isosceles triangle, b α - length of the third side, β And - corresponding angles, R - radius of the circumscribed circle, r

- radius of inscribed .

The sides can be found as follows:

Angles can be expressed in the following ways:

The perimeter of an isosceles triangle can be calculated in any of the following ways:

The area of ​​a triangle can be calculated in one of the following ways:

(Heron's formula).

• Signs
• Two angles of a triangle are equal.
• The height coincides with the median.
• The height coincides with the bisector.
• The bisector coincides with the median.
• The two heights are equal.
• The two medians are equal.

Two bisectors are equal (Steiner-Lemus theorem).

see also

Gremyachinsky municipal district of Perm region

ISOSceles TRIANGLE, A TRIANGLE having two sides of equal length; the angles at these sides are also equal... ISOSCELES TRIANGLE

And (simple) trigon, triangle, man. 1. A geometric figure bounded by three mutually intersecting lines forming three internal angles (mat.). Obtuse triangle. Acute triangle. Right triangle.… … TRIANGLE

ISOSceles, aya, oe: an isosceles triangle having two equal sides. | noun isosceles, and, female Ozhegov's explanatory dictionary. S.I. Ozhegov, N.Yu. Shvedova. 1949 1992 … Ozhegov's Explanatory Dictionary

triangle- ▲ a polygon with three angles, a triangle, the simplest polygon; is defined by 3 points that do not lie on the same line. triangular. acute angle. acute-angled. right triangle: leg. hypotenuse. isosceles triangle. ▼… … Ideographic Dictionary of the Russian Language

triangle- TRIANGLE1, a, m of what or with def. An object in the shape of a geometric figure bounded by three intersecting lines forming three internal angles. She sorted through her husband's letters, yellowed triangles from the front. TRIANGLE2, a, m... ... Explanatory dictionary of Russian nouns

This term has other meanings, see Triangle (meanings). A triangle (in Euclidean space) is a geometric figure formed by three segments that connect three points that do not lie on the same straight line. Three dots,... ... Wikipedia

Triangle (polygon)- Triangles: 1 acute, rectangular and obtuse; 2 regular (equilateral) and isosceles; 3 bisectors; 4 medians and center of gravity; 5 heights; 6 orthocenter; 7 middle line. TRIANGLE, a polygon with 3 sides. Sometimes under... ... Illustrated Encyclopedic Dictionary

encyclopedic Dictionary

triangle- A; m. 1) a) A geometric figure bounded by three intersecting lines forming three internal angles. Rectangular, isosceles triangle. Calculate the area of ​​the triangle. b) ott. what or with def. A figure or object of this shape... ... Dictionary of many expressions

A; m. 1. A geometric figure bounded by three intersecting lines forming three internal angles. Rectangular, isosceles t. Calculate the area of ​​the triangle. // what or with def. A figure or object of this shape. T. roofs. T.… … encyclopedic Dictionary

This lesson will cover the topic “Isosceles triangle and its properties.” You will learn what isosceles and equilateral triangles look like and how they are characterized. Prove the theorem on the equality of angles at the base of an isosceles triangle. Consider also the theorem about the bisector (median and altitude) drawn to the base of an isosceles triangle. At the end of the lesson, you will solve two problems using the definition and properties of an isosceles triangle.

Definition:Isosceles is called a triangle whose two sides are equal.

Rice. 1. Isosceles triangle

AB = AC - sides. BC - foundation.

The area of ​​an isosceles triangle is equal to half the product of its base and height.

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

Rice. 2. Equilateral triangle

AB = BC = SA.

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

Given: AB = AC.

Prove:∠B =∠C.

Rice. 3. Drawing for the theorem

Proof: triangle ABC = triangle ACB according to the first sign (two equal sides and the angle between them). From the equality of triangles it follows that all corresponding elements are equal. This means ∠B = ∠C, which is what needed to be proven.

Theorem 2: In an isosceles triangle bisector drawn to the base is median- length of the third side, height.

Given: AB = AC, ∠1 = ∠2.

Prove:ВD = DC, AD perpendicular to BC.

Rice. 4. Drawing for Theorem 2

Proof: triangle ADB = triangle ADC according to the first sign (AD - general, AB = AC by condition, ∠BAD = ∠DAC). From the equality of triangles it follows that all corresponding elements are equal. BD = DC since they lie opposite equal angles. So AD is the median. Also ∠3 = ∠4, since they lie opposite equal sides. But, besides, they are equal in total. Therefore, ∠3 = ∠4 = . This means that AD is the height of the triangle, which is what we needed to prove.

In the only case a = b = . In this case, the lines AC and BD are called perpendicular.

Since the bisector, height and median are the same segment, the following statements are also true:

The altitude of an isosceles triangle drawn to the base is the median and bisector.

The median of an isosceles triangle drawn to the base is the altitude and bisector.

Example 1: In an isosceles triangle, the base is half the size of the side, and the perimeter is 50 cm. Find the sides of the triangle.

Given: AB = AC, BC = AC. P = 50 cm.

Find: BC, AC, AB.

Solution:

Rice. 5. Drawing for example 1

Let us denote the base BC as a, then AB = AC = 2a.

2a + 2a + a = 50.

5a = 50, a = 10.

Answer: BC = 10 cm, AC = AB = 20 cm.

Example 2: Prove that in an equilateral triangle all angles are equal.

Given: AB = BC = SA.

Prove:∠A = ∠B = ∠C.

Proof:

Rice. 6. Drawing for example

∠B = ∠C, since AB = AC, and ∠A = ∠B, since AC = BC.

Therefore, ∠A = ∠B = ∠C, which is what needed to be proven.

Answer: Proven.

In today's lesson we looked at an isosceles triangle and studied its basic properties. In the next lesson we will solve problems on the topic of isosceles triangles, on calculating the area of ​​an isosceles and equilateral triangle.

1. Alexandrov A.D., Werner A.L., Ryzhik V.I. and others. Geometry 7. - M.: Education.
2. Atanasyan L.S., Butuzov V.F., Kadomtsev S.B. and others. Geometry 7. 5th ed. - M.: Enlightenment.
3. Butuzov V.F., Kadomtsev S.B., Prasolova V.V. Geometry 7 / V.F. Butuzov, S.B. Kadomtsev, V.V. Prasolova, ed. Sadovnichego V.A. - M.: Education, 2010.

1. Dictionaries and encyclopedias on Academician ().
2. Festival of pedagogical ideas “Open Lesson” ().
3. 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. Sadovnichego V.A. - M.: Education, 2010.

2. The perimeter of an isosceles triangle is 35 cm, and the base is three times smaller than the side. Find the sides of the triangle.

3. Given: AB = BC. Prove that ∠1 = ∠2.

4. The perimeter of an isosceles triangle is 20 cm, one of its sides is twice as large as the other. Find the sides of the triangle. How many solutions does the problem have?