When solving geometric problems, it is useful to follow this algorithm. While reading the task statement, it is necessary

• Make a drawing. The drawing should correspond to the condition of the problem as much as possible, so its main task is to help find the solution
• Apply all the data from the task condition to the drawing
• Write out all the geometric concepts that occur in the problem
• Recall all the theorems that relate to this concept
• Put on the drawing all the relationships between the elements of a geometric figure that follow from these theorems

For example, if the task contains the words bisector of the angle of a triangle, you need to remember the definition and properties of the bisector and designate equal or proportional segments and angles in the drawing.

In this article, you will find the basic properties of a triangle that you need to know to successfully solve problems.

TRIANGLE.

Area of ​​a triangle.

1. ,

here - an arbitrary side of the triangle, - the height lowered to this side.

2. ,

here and are arbitrary sides of the triangle, is the angle between these sides:

3. Heron formula:

Here - the lengths of the sides of the triangle, - the semiperimeter of the triangle,

4. ,

here - the semiperimeter of the triangle, - the radius of the inscribed circle.

Let be the lengths of the tangent segments.

Then Heron's formula can be written in the following form:

5.

6. ,

here - the lengths of the sides of the triangle, - the radius of the circumscribed circle.

If a point is taken on a side of a triangle that divides this side in the ratio m:n, then the segment connecting this point with the vertex of the opposite angle divides the triangle into two triangles, the areas of which are related as m:n:

The ratio of the areas of similar triangles is equal to the square of the similarity coefficient.

Triangle median

This is a line segment that connects the vertex of the triangle with the midpoint of the opposite side.

Triangle medians intersect at one point and share the intersection point in a ratio of 2:1, counting from the top.

Intersection point of medians right triangle divides the median into two segments, the smaller of which is equal to the radius of the inscribed circle, and the larger one is equal to the radius of the circumscribed circle.

The radius of the circumscribed circle is twice the radius of the inscribed circle: R=2r

Median length arbitrary triangle

,

here - the median drawn to the side - the lengths of the sides of the triangle.

Bisector of a triangle

This is a segment of the bisector of any angle of a triangle, connecting the vertex of this angle with the opposite side.

Bisector of a triangle divides the side into segments proportional to the adjacent sides:

Triangle bisectors intersect at one point, which is the center of the inscribed circle.

All points on the bisector of an angle are equidistant from the sides of the angle.

Triangle Height

This is a segment of the perpendicular, lowered from the vertex of the triangle to the opposite side, or its continuation. In an obtuse triangle, the altitude drawn from the vertex of an acute angle lies outside the triangle.

The heights of a triangle intersect at one point, which is called the triangle's orthocenter.

To find the height of a triangle drawn to the side, you need to find its area in any way possible, and then use the formula:

Center of a circle circumscribed about a triangle, lies at the point of intersection of the perpendicular bisectors drawn to the sides of the triangle.

The radius of the circumscribed circle of a triangle can be found using the following formulas:

Here, are the lengths of the sides of the triangle, and is the area of ​​the triangle.

,

where is the length of the side of the triangle, is the opposite angle. (This formula follows from the sine theorem).

triangle inequality

Each side of the triangle is less than the sum and greater than the difference of the other two.

The sum of the lengths of any two sides is always greater than the length of the third side:

Opposite the larger side lies a larger angle; opposite the larger angle lies the larger side:

If , then vice versa.

Sine theorem:

The sides of a triangle are proportional to the sines of the opposite angles:

Cosine theorem:

square side of a triangle is equal to the sum squares of the other two sides without doubling the product of these sides by the cosine of the angle between them:

Right triangle

- It is a triangle with one of the angles equal to 90°.

The sum of the acute angles of a right triangle is 90°.

The hypotenuse is the side that lies opposite the 90° angle. The hypotenuse is the longest side.

Pythagorean theorem:

the square of the hypotenuse is equal to the sum of the squares of the legs:

The radius of a circle inscribed in a right triangle is

,

here - the radius of the inscribed circle, - the legs, - the hypotenuse:

Center of a circle circumscribed about a right triangle lies in the middle of the hypotenuse:

Median of a right triangle drawn to the hypotenuse equal to half of the hypotenuse.

Definition of sine, cosine, tangent and cotangent of a right triangle see

The ratio of elements in a right triangle:

The square of the height of a right triangle drawn from the vertex of the right angle is equal to the product of the projections of the legs to the hypotenuse:

The square of the leg is equal to the product of the hypotenuse and the projection of the leg to the hypotenuse:

Leg lying against the corner equal to half the hypotenuse:

Isosceles triangle.

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

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

Top angle.

I - sides

And - angles at the base.

Height, bisector and median.

Attention! The height, bisector and median drawn to the lateral side do not match.

right triangle

(or equilateral triangle ) is a triangle, all sides and angles of which are equal to each other.

Area of ​​an equilateral triangle is equal to

where is the length of the side of the triangle.

Center of a circle inscribed in an equilateral triangle, coincides with the center of the circle circumscribed about an equilateral triangle and lies at the point of intersection of the medians.

Intersection point of medians of an equilateral triangle divides the median into two segments, the smaller of which is equal to the radius of the inscribed circle, and the larger one is equal to the radius of the circumscribed circle.

If one of the angles of an isosceles triangle is 60°, then the triangle is regular.

Middle line of the triangle

This is a segment that connects the midpoints of two sides.

In the figure DE - middle line triangle ABC.

The midline of the triangle is parallel to the third side and equal to half of it: DE||AC, AC=2DE

External corner of a triangle

This is the angle adjacent to any angle of the triangle.

An exterior angle of a triangle is equal to the sum of two angles not adjacent to it.

Trigonometric functions of an external angle:

Signs of equality of triangles:

1 . If two sides and the angle between them of one triangle are respectively equal to two sides and the angle between them of another triangle, then such triangles are congruent.

2 . If a side and two adjacent angles of one triangle are respectively equal to a side and two adjacent angles of another triangle, then such triangles are congruent.

3 If three sides of one triangle are respectively equal to three sides of another triangle, then such triangles are congruent.

Important: since in a right triangle two angles are obviously equal, then for equality of two right triangles only two elements are required to be equal: two sides, or a side and an acute angle.

Signs of similarity of triangles:

1 . If two sides of one triangle are proportional to two sides of another triangle, and the angles enclosed between these sides are equal, then these triangles are similar.

2 . If three sides of one triangle are proportional to three sides of another triangle, then these triangles are similar.

3 . If two angles of one triangle are equal to two angles of another triangle, then these triangles are similar.

Important: In similar triangles, similar sides lie opposite equal angles.

Theorem of Menelaus

Let the line intersect the triangle , where is the point of its intersection with the side , is the point of its intersection with the side , and is the point of its intersection with the extension of the side . Then

Triangles.

Basic concepts.

Triangle- this is a figure consisting of three segments and three points that do not lie on one straight line.

The segments are called parties, and the points peaks.

Sum of angles triangle is equal to 180 º.

The height of the triangle.

Triangle Height is a perpendicular drawn from a vertex to the opposite side.

In an acute-angled triangle, the height is contained inside the triangle (Fig. 1).

In a right triangle, the legs are the heights of the triangle (Fig. 2).

In an obtuse triangle, the height passes outside the triangle (Fig. 3).

Triangle height properties:

Bisector of a triangle.

Bisector of a triangle- this is a segment that bisects the corner of the vertex and connects the vertex to a point on the opposite side (Fig. 5).

Bisector properties:

The median of a triangle.

Triangle median- this is a segment connecting the vertex with the middle of the opposite side (Fig. 9a).

The length of the median can be calculated using the formula: 2b 2 + 2c 2 - a 2 m a 2 = —————— 4 where m a- median drawn to the side a. In a right triangle, the median drawn to the hypotenuse is half the hypotenuse: c mc = — 2 where mc is the median drawn to the hypotenuse c(Fig. 9c) The medians of a triangle intersect at one point (at the center of mass of the triangle) and are divided by this point in a ratio of 2:1, counting from the top. That is, the segment from the vertex to the center is twice the segment from the center to the side of the triangle (Fig. 9c). The three medians of a triangle divide it into six triangles of equal area.

The middle line of the triangle.

Middle line of the triangle- this is a segment connecting the midpoints of its two sides (Fig. 10).

The midline of a triangle is parallel to the third side and equal to half of it.

The outer corner of the triangle.

outside corner triangle is equal to the sum of two non-adjacent interior angles (Fig. 11).

The exterior angle of a triangle is greater than any non-adjacent angle.

Right triangle.

Right triangle- this is a triangle that has a right angle (Fig. 12).

The side of a right triangle opposite the right angle is called hypotenuse.

The other two sides are called legs.

Proportional segments in a right triangle.

1) In a right triangle, the height drawn from the right angle forms three similar triangles: ABC, ACH and HCB (Fig. 14a). Accordingly, the angles formed by the height are equal to the angles A and B.

Fig.14a

Isosceles triangle.

Isosceles triangle- this is a triangle in which two sides are equal (Fig. 13).

These equal sides are called sides, and the third basis triangle.

In an isosceles triangle, the angles at the base are equal. (In our triangle, angle A equal to the angle C).

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

Equilateral triangle.

An equilateral triangle is a triangle in which all sides are equal (Fig. 14).

Properties of an equilateral triangle:

Remarkable properties of triangles.

Triangles have original properties that will help you successfully solve problems associated with these shapes. Some of these properties are outlined above. But we repeat them again, adding a few other great features to them:

1) In a right triangle with angles 90º, 30º and 60º, the leg b, lying opposite the angle of 30º, is equal to half of the hypotenuse. A lega more legb√3 times (Fig. 15 a). For example, if the leg of b is 5, then the hypotenuse c necessarily equal to 10, and the leg a equals 5√3. 2) In a right-angled isosceles triangle with angles of 90º, 45º and 45º, the hypotenuse is √2 times the leg (Fig. 15 b). For example, if the legs are 5, then the hypotenuse is 5√2. 3) The middle line of the triangle is equal to half of the parallel side (Fig. 15 With). For example, if the side of a triangle is 10, then the midline parallel to it is 5. 4) In a right triangle, the median drawn to the hypotenuse is equal to half of the hypotenuse (Fig. 9c): mc= c/2. 5) The medians of a triangle, intersecting at one point, are divided by this point in a ratio of 2:1. That is, the segment from the vertex to the point of intersection of the medians is twice the segment from the point of intersection of the medians to the side of the triangle (Fig. 9c) 6) In a right triangle, the midpoint of the hypotenuse is the center of the circumscribed circle (Fig. 15 d).

Signs of equality of triangles.

The first sign of equality: If two sides and the angle between them of one triangle are equal to two sides and the angle between them of another triangle, then such triangles are congruent.

The second sign of equality: if the side and angles adjacent to it of one triangle are equal to the side and angles adjacent to it of another triangle, then such triangles are congruent.

The third sign of equality: If three sides of one triangle are equal to three sides of another triangle, then such triangles are congruent.

Triangle Inequality.

In any triangle, each side is less than the sum of the other two sides.

Pythagorean theorem.

In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs:

c 2 = a 2 + b 2 .

Area of ​​a triangle.

1) The area of ​​a triangle is equal to half the product of its side and the height drawn to this side:

Ah
S = ——
2

2) The area of ​​a triangle is equal to half the product of any two of its sides and the sine of the angle between them:

1
S = — AB · AC · sin A
2

A triangle circumscribed about a circle.

A circle is called inscribed in a triangle if it touches all its sides (Fig. 16 a).

Triangle inscribed in a circle.

A triangle is called inscribed in a circle if it touches it with all vertices (Fig. 17 a).

Sine, cosine, tangent, cotangent of an acute angle of a right triangle (Fig. 18).

Sinus acute angle x opposite catheter to the hypotenuse.
Denoted like this: sinx.

Cosine acute angle x right triangle is the ratio adjacent catheter to the hypotenuse.
It is denoted as follows: cos x.

Tangent acute angle x is the ratio of the opposite leg to the adjacent leg.
Denoted like this: tgx.

Cotangent acute angle x is the ratio of the adjacent leg to the opposite leg.
Denoted like this: ctgx.

Rules:

Leg opposite corner x, is equal to the product of the hypotenuse and sin x:

b=c sin x

Leg adjacent to the corner x, is equal to the product of the hypotenuse and cos x:

a = c cos x

Leg opposite corner x, is equal to the product of the second leg and tg x:

b = a tg x

Leg adjacent to the corner x, is equal to the product of the second leg and ctg x:

a = b ctg x.

For any acute angle x:

sin (90° - x) = cos x

cos (90° - x) = sin x

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In fact, everything is not so scary at all. Of course, the "real" definition of sine, cosine, tangent and cotangent should be looked at in the article. But you really don't want to, do you? We can rejoice: to solve problems about a right triangle, you can simply fill in the following simple things:

What about the angle? Is there a leg that is opposite the corner, that is, the opposite leg (for the corner)? Of course have! This is a cathet!

But what about the angle? Look closely. Which leg is adjacent to the corner? Of course, the cat. So, for the angle, the leg is adjacent, and

And now, attention! Look what we got:

See how great it is:

Now let's move on to tangent and cotangent.

How to put it into words now? What is the leg in relation to the corner? Opposite, of course - it "lies" opposite the corner. And the cathet? Adjacent to the corner. So what did we get?

See how the numerator and denominator are reversed?

And now again the corners and made the exchange:

Summary

Let's briefly write down what we have learned.

Pythagorean theorem:

The main right triangle theorem is the Pythagorean theorem.

Pythagorean theorem

By the way, do you remember well what the legs and hypotenuse are? If not, then look at the picture - refresh your knowledge

It is possible that you have already used the Pythagorean theorem many times, but have you ever wondered why such a theorem is true. How would you prove it? Let's do like the ancient Greeks. Let's draw a square with a side.

You see how cunningly we divided its sides into segments of lengths and!

Now let's connect the marked points

Here we, however, noted something else, but you yourself look at the picture and think about why.

What is the area of ​​the larger square?

Right, .

What about the smaller area?

Certainly, .

The total area of ​​the four corners remains. Imagine that we took two of them and leaned against each other with hypotenuses.

What happened? Two rectangles. So, the area of ​​"cuttings" is equal.

Let's put it all together now.

Let's transform:

So we visited Pythagoras - we proved his theorem in an ancient way.

Right triangle and trigonometry

For a right triangle, the following relations hold:

The sine of an acute angle is equal to the ratio of the opposite leg to the hypotenuse

The cosine of an acute angle is equal to the ratio of the adjacent leg to the hypotenuse.

The tangent of an acute angle is equal to the ratio of the opposite leg to the adjacent leg.

The cotangent of an acute angle is equal to the ratio of the adjacent leg to the opposite leg.

And once again, all this in the form of a plate:

It is very comfortable!

Signs of equality of right triangles

I. On two legs

II. By leg and hypotenuse

III. By hypotenuse and acute angle

IV. Along the leg and acute angle

a)

b)

Attention! Here it is very important that the legs are "corresponding". For example, if it goes like this:

THEN THE TRIANGLES ARE NOT EQUAL, despite the fact that they have one identical acute angle.

Need to in both triangles the leg was adjacent, or in both - opposite.

Have you noticed how the signs of equality of right triangles differ from the usual signs of equality of triangles?

Look at the topic “and pay attention to the fact that for the equality of “ordinary” triangles, you need the equality of their three elements: two sides and an angle between them, two angles and a side between them, or three sides.

But for the equality of right-angled triangles, only two corresponding elements are enough. It's great, right?

Approximately the same situation with signs of similarity of right triangles.

Signs of similarity of right triangles

I. Acute corner

II. On two legs

III. By leg and hypotenuse

Median in a right triangle

Why is it so?

Consider a whole rectangle instead of a right triangle.

Let's draw a diagonal and consider a point - the point of intersection of the diagonals. What do you know about the diagonals of a rectangle?

And what follows from this?

So it happened that

1. - median:

Remember this fact! Helps a lot!

What is even more surprising is that the converse is also true.

What good can be gained from the fact that the median drawn to the hypotenuse is equal to half the hypotenuse? Let's look at the picture

Look closely. We have: , that is, the distances from the point to all three vertices of the triangle turned out to be equal. But in a triangle there is only one point, the distances from which about all three vertices of the triangle are equal, and this is the CENTER OF THE CIRCUM DEscribed. So what happened?

So let's start with this "besides...".

Let's look at i.

But in similar triangles all angles are equal!

The same can be said about and

Now let's draw it together:

What use can be drawn from this "triple" similarity.

Well, for example - two formulas for the height of a right triangle.

We write the relations of the corresponding parties:

To find the height, we solve the proportion and get first formula "Height in a right triangle":

Well, now, applying and combining this knowledge with others, you will solve any problem with a right triangle!

So, let's apply the similarity: .

What will happen now?

Again we solve the proportion and get the second formula:

Both of these formulas must be remembered very well and the one that is more convenient to apply.

Let's write them down again.

Pythagorean theorem:

In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs:.

Signs of equality of right triangles:

• on two legs:
• along the leg and hypotenuse: or
• along the leg and the adjacent acute angle: or
• along the leg and the opposite acute angle: or
• by hypotenuse and acute angle: or.

Signs of similarity of right triangles:

• one sharp corner: or
• from the proportionality of the two legs:
• from the proportionality of the leg and hypotenuse: or.

Sine, cosine, tangent, cotangent in a right triangle

• The sine of an acute angle of a right triangle is the ratio of the opposite leg to the hypotenuse:
• The cosine of an acute angle of a right triangle is the ratio of the adjacent leg to the hypotenuse:
• The tangent of an acute angle of a right triangle is the ratio of the opposite leg to the adjacent one:
• The cotangent of an acute angle of a right triangle is the ratio of the adjacent leg to the opposite:.

Height of a right triangle: or.

In a right triangle, the median drawn from the vertex of the right angle is equal to half the hypotenuse: .

Area of ​​a right triangle:

• through the catheters: