An isosceles triangle and its sides. Features that make up the elements and properties of an isosceles triangle

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

Properties

Angles opposite to equal sides of an isosceles triangle are equal to each other. The bisectors, medians and heights drawn from these angles are also equal.
The bisector, median, height and perpendicular to the base coincide. The centers of the inscribed and circumscribed circles lie on this line.
Angles opposite to equal sides are always sharp (follows from their equality).

Let be a- the length of two equal sides of an isosceles triangle, b- the length of the third side, α and β - the corresponding angles, R- the radius of the circumscribed circle, r- inscribed radius.

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

The two corners of the triangle are equal.
The height matches 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).

Isosceles triangle properties

The properties of an isosceles triangle are expressed in terms of 5 theorems:

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

Proof of the theorem:

Consider an isosceles Δ ABC with the foundation AS .

The sides are equal AB = Sun ,

Therefore, the angles at the base ∠ BАC = ∠ BCA .

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

Theorem 2. In an isosceles triangle, the bisector drawn 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 median.

Proof of the theorem:

Dan Δ ABC .
From point V let's hold the height BD.
The triangle is divided into Δ ABD and Δ CBD. These triangles are equal because their hypotenuse and common leg are equal ().
Direct AS and BD are called perpendicular.
B Δ ABD and Δ BCD ∠ BAD = ∠ BСD (from Theorem 1).
AB = BC - the sides are equal.
Parties AD = CD, since point D divides the segment in half.
Hence Δ ABD = Δ BCD.
The bisector, height and median are one segment - BD

Output:

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

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

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

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 in the first attribute).
Let Δ A 1 B 1 C 2 = Δ ABC, whose vertex C 2 lies in the same half-plane with 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. Hence, the lines A 1 D and B 1 D are perpendicular to the line C 1 C 2. A 1 D and B 1 D have different points A 1 and B 1, therefore, do not coincide. But through point D of straight line C 1 C 2 only one straight line perpendicular to it can be drawn.
From here we came to a contradiction and proved the theorem.

Signs of an isosceles triangle

If two angles in a triangle are equal.
The sum of the angles of a triangle is 180 °.
If in a triangle, the bisector is the median or height.
If in a triangle, the median is the bisector or height.
If in a triangle, the height is the median or bisector.

Isosceles triangle formulas

b- side (base)
a- equal sides
a - angles at the base
b

Side length formulas(grounds - b):

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

Equal side length formulas - (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
b- side (base)
a- equal sides
a - angles at the base
b - the angle formed by equal sides

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

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

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

L = \ sqrt (a ^ (2) -b ^ (2) / 4)

b- side (base)
a- equal sides
h- height

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

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

This lesson will consider the topic "isosceles triangle and its properties." You will learn what the isosceles and equilateral triangles look like and are characterized by. Prove the theorem on the equality of angles at the base of an isosceles triangle. Consider also the theorem on the bisector (median and height) drawn to the base of an isosceles triangle. At the end of the lesson, you will break down two problems using the definition and properties of an isosceles triangle.

Definition:Isosceles called a triangle with two sides equal.

Rice. 1. Isosceles triangle

AB = AC - lateral sides. BC is the base.

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

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

Rice. 2. Equilateral triangle

AB = BC = CA.

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

Given: AB = AC.

Prove:∠В = ∠С.

Rice. 3. Drawing to the theorem

Proof: triangle ABC = triangle ACB on the first basis (on two equal sides and the angle between them). Equality of triangles implies equality of all corresponding elements. Hence, ∠В = ∠С, as required.

Theorem 2: In an isosceles triangle bisector taken to the base is median and height.

Given: AB = AC, ∠1 = ∠2.

Prove: BD = DC, AD perpendicular to BC.

Rice. 4. Drawing to Theorem 2

Proof: triangle ADB = triangle ADC by the first attribute (AD - common, AB = AC by condition, ∠BAD = ∠DAC). Equality of triangles implies equality of all corresponding elements. BD = DC since they are opposite equal angles. This means AD is the median. Also ∠3 = ∠4, since they are opposite to equal sides. But, besides, they add up. Therefore, ∠3 = ∠4 =. Hence, AD is the height of the triangle, as required.

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

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

The height 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 height and bisector.

Example 1: In an isosceles triangle, the base is half 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's designate 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 all angles are equal in an equilateral triangle.

Given: AB = BC = CA.

Prove:∠А = ∠В = ∠С.

Proof:

Rice. 6. Drawing for example

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

Therefore, ∠A = ∠B = ∠C, as required.

Answer: Proven.

In today's lesson, we examined an isosceles triangle, studied its basic properties. In the next lesson, we will solve problems on the topic of an isosceles triangle, to calculate the areas of an isosceles and equilateral triangle.

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

Dictionaries and encyclopedias on "Academician" ().
Festival of Pedagogical Ideas "Open Lesson" ().
Кaknauchit.ru ().

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

2. The perimeter of an isosceles triangle is 35 cm, and the base is three times less than the lateral 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?

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.

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 - a part of the 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

An ordinary versatile triangle is a closed geometric figure consisting of three segments of different lengths and three angles, none of which is right. 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.

An isosceles triangle and its sides. Features that make up the elements and properties of an isosceles triangle

Properties

Signs

see also

See what "isosceles triangle" is in other dictionaries:

Isosceles triangle properties

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

Signs of an isosceles triangle

Isosceles triangle formulas

What are triangles

Distinctive features

Geometric properties of the figure

Height of an isosceles triangle

Median of an isosceles triangle

How to determine the perimeter

Area of an isosceles triangle

Other properties of an isosceles triangle

An isosceles triangle and its sides. Features that make up the elements and properties of an isosceles triangle

Properties

Signs

see also

See what "isosceles triangle" is in other dictionaries:

Isosceles triangle properties

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

Signs of an isosceles triangle

Isosceles triangle formulas

What are triangles

Distinctive features

Geometric properties of the figure

Height of an isosceles triangle

Median of an isosceles triangle

How to determine the perimeter

Area of ​​an isosceles triangle

Other properties of an isosceles triangle

Area of an isosceles triangle