Isosceles acute triangle. 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 Unified State Exam problems 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 the parties. 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 given isosceles triangular triangle, feel free to draw the height, get two right triangles and solve the problem about 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:

Here we go 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 table values 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. This happens very often: topics , “ Isosceles triangle”and in problems they go together in groups, 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 - sides, is the base of an 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 this bisector and median.

Let's summarize:

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

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

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.
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:

(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).

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 midline. 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. Consider an isosceles triangle ABC with base BC and prove that ∠ B = ∠ C. Let AD be the bisector 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).

Let's consider arbitrary point M, lying on a straight river. 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 accordingly 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 external corner at the vertex C is 123°. Find the size of angle ABC. Give your answer in degrees.

Video solution.

Definition 7. Any triangle whose two sides are equal is called isosceles.
Two equal sides are called lateral, the third is called the base.
Definition 8. If all three sides of a triangle are equal, then the triangle is called equilateral.
It is a special type of isosceles triangle.
Theorem 18. The height of an isosceles triangle, lowered to the base, is at the same time the bisector of the angle between equal sides, the median and the axis of symmetry of the base.
Proof. Let us lower the height to the base of the isosceles triangle. It will divide it into two equal (along the leg and hypotenuse) right triangles. Angles A and C are equal, and the height also divides the base in half and will be the axis of symmetry of the entire figure under consideration.
This theorem can also be formulated as follows:
Theorem 18.1. The median of an isosceles triangle, lowered to the base, is also the bisector of the angle between equal sides, the height and the axis of symmetry of the base.
Theorem 18.2. The bisector of an isosceles triangle, lowered to the base, is simultaneously the height, median and axis of symmetry of the base.
Theorem 18.3. The axis of symmetry of an isosceles triangle is simultaneously the bisector of the angle between equal sides, the median and the altitude.
The proof of these consequences also follows from the equality of the triangles into which an isosceles triangle is divided.

Theorem 19. The angles at the base of an isosceles triangle are equal.
Proof. Let us lower the height to the base of the isosceles triangle. She will divide it into two equal (along the leg and hypotenuse) right triangles, which means corresponding angles are equal, i.e. ∠ A=∠ C
The criteria for an isosceles triangle come from Theorem 1 and its corollaries and Theorem 2.
Theorem 20. If two of the indicated four lines (height, median, bisector, axis of symmetry) coincide, then the triangle will be isosceles (which means all four lines will coincide).
Theorem 21. If any two angles of a triangle are equal, then it is isosceles.

Proof: Similar to the proof of the direct theorem, but using the second criterion for the equality of triangles. The center of gravity, the centers of the circumcircle and incircle, and the point of intersection of the altitudes of an isosceles triangle all lie on its axis of symmetry, i.e. on top.
An equilateral triangle is isosceles for each pair of its sides. Due to the equality of all its sides, all three angles of such a triangle are equal. Considering that the sum of the angles of any triangle is equal to two right angles, we see that each of the angles equilateral triangle equals 60°. Conversely, to ensure that all sides of a triangle are equal, it is enough to check that two of its three angles are equal to 60°.
Theorem 22 . In an equilateral triangle, all the remarkable points coincide: the center of gravity, the centers of the inscribed and circumscribed circles, the point of intersection of the altitudes (called the orthocenter of the triangle).
Theorem 23 . If two of the indicated four points coincide, then the triangle will be equilateral and, as a consequence, all four named points will coincide.
Indeed, such a triangle will turn out, according to the previous one, isosceles with respect to any pair of sides, i.e. equilateral. An equilateral triangle is also called a regular triangle. The area of an isosceles triangle is equal to half the product of the square of the side side and the sine of the angle between the sides
Consider this formula for an equilateral triangle, then the alpha angle will be equal to 60 degrees. Then the formula will change to this:

Theorem d1 . In an isosceles triangle, the medians drawn to the sides are equal.

Proof: Let ABC be an isosceles triangle (AC = BC), AK and BL its medians. Then triangles AKB and ALB are equal according to the second criterion for the equality of triangles. They have common side AB, sides AL and BK are equal as halves of the lateral sides of an isosceles triangle, and angles LAB and KBA are equal as the base angles of an isosceles triangle. Since the triangles are congruent, their sides AK and LB are equal. But AK and LB are the medians of an isosceles triangle drawn to its lateral sides.
Theorem d2 . In an isosceles triangle, the bisectors drawn to the lateral sides are equal.

Proof: Let ABC be an isosceles triangle (AC = BC), AK and BL its bisectors. Triangles AKB and ALB are equal according to the second criterion for the equality of triangles. They have a common side AB, angles LAB and KBA are equal as the base angles of an isosceles triangle, and angles LBA and KAB are equal as half the base angles of an isosceles triangle. Since the triangles are congruent, their sides AK and LB - the bisectors of triangle ABC - are congruent. The theorem has been proven.
Theorem d3 . In an isosceles triangle, the heights at the sides are equal.

Proof: Let ABC be an isosceles triangle (AC = BC), AK and BL its altitudes. Then angles ABL and KAB are equal, since angles ALB and AKB are right angles, and angles LAB and ABK are equal as the base angles of an isosceles triangle. Consequently, triangles ALB and AKB are equal according to the second criterion for the equality of triangles: they have a common side AB, angles KAB and LBA are equal according to the above, and angles LAB and KBA are equal as the base angles of an isosceles triangle. If the triangles are congruent, their sides AK and BL are also congruent. Q.E.D.