In an isosceles triangle, two sides are equal. Isosceles triangle

Lesson topic

Isosceles triangle

The purpose of the lesson

Introduce students to an isosceles triangle;
Continue to develop skills in constructing right triangles;
Expand schoolchildren’s knowledge about the properties of isosceles triangles;
Strengthen theoretical knowledge when solving problems.

Lesson Objectives

Be able to formulate, prove and use the theorem on the properties of an isosceles triangle in the process of solving problems;
Continue the development of conscious perception of educational material, logical thinking, self-control and self-esteem skills;
Arouse cognitive interest in mathematics lessons;
Foster activity, curiosity and organization.

Lesson Plan

1. General concepts and definitions about an isosceles triangle.
2. Properties of an isosceles triangle.
3. Signs of an isosceles triangle.
4. Questions and tasks.

Isosceles triangle

An isosceles triangle is a triangle that has two equal sides, called the sides of an isosceles triangle, and its third side is called the base.

The top of a given figure is the one located opposite its base.

The angle that lies opposite the base is called the vertex angle of this triangle, and the other two angles are called the base angles of an isosceles triangle.

Types of isosceles triangles

An isosceles triangle, like other figures, can have different types. Among isosceles triangles there are acute, rectangular, obtuse and equilateral triangles.

An acute triangle has all acute angles.
A right triangle has a straight angle at its apex and sharp angles at its base.
An obtuse angle has an obtuse angle at the apex, and the angles at its base are acute.
An equilateral object has all its angles and sides equal.

Properties of an isosceles triangle

Opposite angles in relation to equal sides of an isosceles triangle are equal to each other;

Bisectors, medians and altitudes drawn from angles opposite equal sides of a triangle are equal to each other.

The bisector, median and height, directed and drawn to the base of the triangle, coincide with each other.

The centers of the inscribed and circumscribed circles lie at the altitude, bisector and median (they coincide) drawn to the base.

Angles opposite equal sides of an isosceles triangle are always acute.

These properties of an isosceles triangle are used in solving problems.

Homework

1. Define an isosceles triangle.
2. What is special about this triangle?
3. How does an isosceles triangle differ from a right triangle?
4. Name the properties of an isosceles triangle that you know.
5. Do you think it is possible in practice to check the equality of angles at the base and how to do this?

Exercise

Now let's conduct a short survey and find out how you learned the new material.

Listen carefully to the questions and answer whether the following statement is true:

1. Can a triangle be considered isosceles if its two sides are equal?
2. A bisector is a segment that connects the vertex of a triangle with the midpoint of the opposite side?
3. A bisector is a segment that bisects an angle that connects a vertex with a point on the opposite side?

Tips for solving isosceles triangle problems:

1. To determine the perimeter of an isosceles triangle, it is enough to multiply the length of the side by 2 and add this product with the length of the base of the triangle.
2. If the perimeter and length of the base of an isosceles triangle are known in the problem, then to find the length of the side it is enough to subtract the length of the base from the perimeter and divide the found difference by 2.
3. And to find the length of the base of an isosceles triangle, knowing both the perimeter and the length of the side, you just need to multiply the side by two and subtract this product from the perimeter of our triangle.

Tasks:

1. Among the triangles in the figure, identify one extra one and explain your choice:

2. Determine which of the triangles shown in the figure are isosceles, name their bases and sides, and also calculate their perimeter.

3. The perimeter of an isosceles triangle is 21 cm. Find the sides of this triangle if one of them is 3 cm larger. How many solutions can this problem have?

4. It is known that if the lateral side and the angle opposite to the base of one isosceles triangle are equal to the lateral side and the angle of another, then these triangles will be equal. Prove this statement.

5. Think and say whether any isosceles triangle is equilateral? And will any equilateral triangle be isosceles?

6. If the sides of an isosceles triangle are 4 m and 5 m, then what will be its perimeter? How many solutions can this problem have?

7. If one of the angles of an isosceles triangle is equal to 91 degrees, then what are the other angles equal to?

8. Think and answer, what angles should a triangle have in order for it to be both rectangular and isosceles?

How many of you know what Pascal's triangle is? The problem of constructing Pascal's triangle is often asked to test basic programming skills. In general, Pascal's triangle relates to combinatorics and probability theory. So what kind of triangle is this?

Pascal's triangle is an infinite arithmetic triangle or triangle-shaped table that is formed using binomial coefficients. In simple words, the vertex and sides of this triangle are ones, and it itself is filled with the sums of the two numbers that are located above. Such a triangle can be folded ad infinitum, but if we outline it, we will get an isosceles triangle with symmetrical lines relative to its vertical axis.

Think about it, where in everyday life have you come across isosceles triangles? Isn’t it true that the roofs of houses and ancient architectural structures are very reminiscent of them? Do you remember what the basis of the Egyptian pyramids is? Where else have you come across isosceles triangles?

Since ancient times, isosceles triangles have helped the Greeks and Egyptians in determining distances and heights. For example, the ancient Greeks used it to determine from afar the distance to a ship at sea. And the ancient Egyptians determined the height of their pyramids based on the length of the cast shadow, because... it was an isosceles triangle.

Since ancient times, people already appreciated the beauty and practicality of this figure, since the shapes of triangles surround us everywhere. Moving through different villages, we see the roofs of houses and other buildings that remind us of an isosceles triangle; when we go into a store, we see triangular-shaped packages of food and juices, and even some human faces have the shape of a triangle. This figure is so popular that you can see it at every step.

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- length of the third side, α And β - 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:

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

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See what an “Isosceles triangle” is in other dictionaries:

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

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

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

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.

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.

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.

Let a- the length of two equal sides of an isosceles triangle, b- length of the third side, h- height of an isosceles triangle

• $a\; =\; \backslash frac\; b\; (2\; \backslash cos\; \backslash alpha)$(a corollary of the cosine theorem);
• $b\; =\; a\; \backslash sqrt\; (2\; (1\; -\; \backslash cos\; \backslash beta))$(a corollary 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 $(\backslash pi)$ And $R$. A triangle is divided in half by its median, and received The angles of two equal right triangles are calculated:

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

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 about right triangles are discussed in, 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:

1. If the triangle is isosceles, then the angles at the base are equal, and the altitude, bisector and median drawn to the base coincide.
2. 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:

1. If in a certain triangle two angles are equal, then it is isosceles.
2. 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.