Derivation of the formula for the height of a right-angled triangle. Right triangle. I. On a sharp corner

Property: 1. In any right triangle, the height dropped from the right angle (by the hypotenuse) divides the right triangle into three similar triangles.

Property: 2. The height of a right-angled triangle, dropped on the hypotenuse, is equal to the geometric mean of the projections of the legs to the hypotenuse (or the geometric mean of those segments into which the height breaks the hypotenuse).

Property: 3. The leg is equal to the geometric mean of the hypotenuse and the projection of this leg to the hypotenuse.

Property: 4. The leg against an angle of 30 degrees is equal to half the hypotenuse.

Formula 1.

Formula 2. where is the hypotenuse; , legs.

Property: 5. In a right-angled triangle, the median drawn to the hypotenuse is equal to its half and is equal to the radius of the circumscribed circle.

Property: 6. Dependence between the sides and corners of a right-angled triangle:

44. Theorem of cosines. Consequences: connection between diagonals and sides of a parallelogram; determination of the type of triangle; formula for calculating the length of the median of a triangle; calculating the cosine of the angle of the triangle.

End of work -

This topic belongs to the section:

Class. Colloquium program basics of planimetry

The property of adjacent angles .. determination of two adjacent angles if one side is common to the other two forming a straight line ..

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In fact, it's not that scary at all. Of course, the "real" definitions of sine, cosine, tangent and cotangent should be found in the article. But I really don't want to, right? We can rejoice: to solve problems about a right-angled triangle, you can simply fill in the following simple things:

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

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

Now, attention! Look what we got:

You see how great:

Now let's move on to tangent and cotangent.

How can I write it down in words now? What is the leg in relation to the corner? Opposite, of course - it "lies" opposite the corner. And the leg? Adjacent to the corner. So what did we do?

See the numerator and denominator are reversed?

And now again the corners and made the exchange:

Summary

Let's briefly write down everything we have learned.

Pythagorean theorem:

The main theorem about a right-angled triangle is the Pythagorean theorem.

Pythagorean theorem

By the way, do you remember well what 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 can I prove it? Let's do like the ancient Greeks. Let's draw a square with a side.

You see how cleverly we divided its sides into lengths and!

Now let's connect the marked points

Here we, however, have noted something else, but you yourself look at the drawing and think about why this is so.

What is the area of ​​the larger square?

Right, .

A smaller area?

Certainly, .

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

What happened? Two rectangles. This means that the area of ​​the "scraps" is equal to.

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-angled triangle, the following relationships 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 is in the form of a plate:

It is very comfortable!

Equality tests for right-angled triangles

I. On two legs

II. On the leg and hypotenuse

III. By hypotenuse and acute angle

IV. On a leg and a sharp corner

a)

b)

Attention! It is very important here that the legs are "appropriate". For example, if it is like this:

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

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

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

Take a look at the topic “and pay attention to the fact that for 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. Great, isn't it?

The situation is approximately the same with the signs of the similarity of right-angled triangles.

Signs of the similarity of right-angled triangles

I. On a sharp corner

II. On two legs

III. On the leg and hypotenuse

Median in a right triangle

Why is this so?

Consider a whole rectangle instead of a right-angled triangle.

Let's draw a diagonal and consider a point - the point of intersection of the diagonals. What is known about the diagonals of a rectangle?

And what follows from this?

So it turned out that

1. - median:

Remember this fact! Helps a lot!

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

What good can you get from the fact that the median drawn to the hypotenuse is equal to half of 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 DESCRIBED CIRCLE. So what happened?

Let's start with this "besides ..."

Let's look at and.

But in such triangles all angles are equal!

The same can be said about and

Now let's draw it together:

What benefit can be derived from this "triple" similarity.

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

Let's write down the relationship of the respective parties:

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

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

So, let's apply the similarity:.

What happens now?

Again we solve the proportion and get the second formula:

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

Let's write them down again

Pythagorean theorem:

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

Signs of equality of right-angled triangles:

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

Signs of the similarity of right-angled triangles:

• one sharp corner: or
• from the proportionality of the two legs:
• from the proportionality of the leg and the 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-angled triangle is the ratio of the opposite leg to the adjacent one:
• The cotangent of an acute angle of a right-angled triangle is the ratio of the adjacent leg to the opposite one:.

Height of a right triangle: or.

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

Area of ​​a right triangle:

• through the legs:

(ABC) and its properties, which is shown in the figure. A right-angled triangle has a hypotenuse - the side that lies opposite the right angle.

Tip 1: How to find the height in a right triangle

The sides that form a right angle are called legs. The side picture AD, DC and BD, DC- legs, and sides AS and SV- hypotenuse.

Theorem 1. In a right-angled triangle with an angle of 30 °, the leg opposite to this angle breaks half of the hypotenuse.

hC

AB- hypotenuse;

AD and DB

Triangle
There is a theorem:
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Solution: 1) The diagonals of any rectangle are equal. Correct 2) If there is one acute angle in a triangle, then this triangle is acute. Not true. Types of triangles. A triangle is called acute-angled if all three of its corners are acute, that is, less than 90 ° 3) If the point lies on.

Or, in another entry,

By the Pythagorean theorem

What is the height in a right triangle formula

Height of a right triangle

The height of a right-angled triangle, drawn to the hypotenuse, can be found in one way or another, depending on the data in the problem statement.

Or, in another entry,

Where BK and KC are the projections of the legs to the hypotenuse (the segments into which the height divides the hypotenuse).

The height drawn to the hypotenuse can be found through the area of ​​a right-angled triangle. If we apply the formula to find the area of ​​a triangle

(half of the product of the side by the height drawn to this side) to the hypotenuse and the height drawn to the hypotenuse, we get:

From here we can find the height as the ratio of the doubled area of ​​the triangle to the length of the hypotenuse:

Since the area of ​​a right-angled triangle is half the product of the legs:

That is, the length of the height drawn to the hypotenuse is equal to the ratio of the product of the legs to the hypotenuse. If we denote the lengths of the legs through a and b, the length of the hypotenuse through c, the formula can be rewritten as

Since the radius of a circle circumscribed about a right-angled triangle is equal to half of the hypotenuse, the length of the height can be expressed in terms of the legs and the radius of the circumscribed circle:

Since the height drawn to the hypotenuse forms two more right-angled triangles, its length can be found through the ratios in the right-angled triangle.

From a right-angled triangle ABK

From right triangle ACK

The length of the height of a right-angled triangle can be expressed in terms of the length of the legs. Because

By the Pythagorean theorem

If you square both sides of the equality:

You can get another formula for connecting the height of a right-angled triangle with the legs:

What is the height in a right triangle formula

Right triangle. Average level.

Do you want to test your strength and find out the result of how ready you are for the Unified State Exam or the OGE?

The main theorem about a right-angled triangle is the Pythagorean theorem.

Pythagorean theorem

By the way, do you remember well what 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 can I prove it? Let's do like the ancient Greeks. Let's draw a square with a side.

You see how cleverly we divided its sides into lengths and!

Now let's connect the marked points

Here we, however, have noted something else, but you yourself look at the drawing and think about why this is so.

What is the area of ​​the larger square? Right, . A smaller area? Certainly, . The total area of ​​the four corners remains. Imagine that we took them two at a time and leaned them against each other with hypotenuses. What happened? Two rectangles. This means that the area of ​​the "scraps" is equal to.

Let's put it all together now.

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

Right triangle and trigonometry

For a right-angled triangle, the following relationships 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 is in the form of a plate:

Have you noticed one very convenient thing? Look at the sign carefully.

It is very comfortable!

Equality tests for right-angled triangles

II. On the leg and hypotenuse

III. By hypotenuse and acute angle

IV. On a leg and a sharp corner

Attention! It is very important here that the legs are "appropriate". For example, if it is like this:

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

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

Have you noticed how the signs of equality of right triangles differ from the usual signs of equality of triangles? Take a look at the topic "Triangle" and pay attention to the fact that for 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. Great, isn't it?

The situation is approximately the same with the signs of the similarity of right-angled triangles.

Signs of the similarity of right-angled triangles

III. On the leg and hypotenuse

Median in a right triangle

Consider a whole rectangle instead of a right-angled triangle.

Let's draw a diagonal and consider the point of intersection of the diagonals. What is known about the diagonals of a rectangle?

The intersection point of the diagonal is halved. The diagonals are equal to

And what follows from this?

So it turned out that

Remember this fact! Helps a lot!

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

What good can you get from the fact that the median drawn to the hypotenuse is equal to half of 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 DESCRIBED CIRCLE. So what happened?

Let's start with this “besides. ".

But in such triangles all angles are equal!

The same can be said about and

Now let's draw it together:

Have the same sharp corners!

What benefit can be derived from this "triple" similarity.

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

Let's write down the relationship of the respective parties:

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

How do you get a second one?

Now let's apply the similarity of triangles and.

So, let's apply the similarity:.

What happens now?

Again we solve the proportion and get the second formula "Height in a right triangle":

Both of these formulas must be very well remembered and whichever is more convenient to apply. Let's write them down again

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

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The property of the height of a right-angled triangle dropped by the hypotenuse

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Right triangle properties

Consider a right triangle (ABC) and its properties, which is shown in the figure. A right-angled triangle has a hypotenuse - the side that lies opposite the right angle. The sides that form a right angle are called legs. The side picture AD, DC and BD, DC- legs, and sides AS and SV- hypotenuse.

Signs of equality of a right-angled triangle:

Theorem 1. If the hypotenuse and leg of a right triangle are similar to the hypotenuse and leg of another triangle, then such triangles are equal.

Theorem 2. If two legs of a right triangle are equal to two legs of another triangle, then such triangles are equal.

Theorem 3. If the hypotenuse and an acute angle of a right triangle are similar to the hypotenuse and an acute angle of another triangle, then such triangles are equal.

Theorem 4. If a leg and an adjacent (opposite) acute angle of a right triangle are equal to a leg and an adjacent (opposite) acute angle of another triangle, then such triangles are equal.

Properties of the leg opposite to the 30 ° angle:

Theorem 1.

Height in a right triangle

In a right-angled triangle with an angle of 30 °, the leg opposite to this angle breaks to half of the hypotenuse.

Theorem 2. If in a right-angled triangle the leg is equal to half of the hypotenuse, then the opposite angle is 30 °.

If the height is drawn from the apex of a right angle to the hypotenuse, then such a triangle is divided into two smaller ones, similar to the outgoing and similar to each other. This leads to the following conclusions:

1. Height is the geometric mean (proportional mean) of the two hypotenuse segments.
2. Each leg of the triangle is the average proportional to the hypotenuse and adjacent segments.

In a right-angled triangle, the legs act as heights. The orthocenter is the point at which the triangle heights intersect. It coincides with the vertex of the right corner of the shape.

hC- the height outgoing from the right angle of the triangle;

AB- hypotenuse;

AD and DB- the segments that arose when dividing the hypotenuse by height.

Return to viewing references for the discipline "Geometry"

Triangle- it geometric figure consisting of three points (vertices) that are not on the same straight line and three line segments connecting these points. A right-angled triangle is a triangle that has one of the angles of 90 ° (right angle).
There is a theorem: the sum of the acute angles of a right-angled triangle is 90 °.
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Keywords: triangle, right-angled, leg, hypotenuse, Pythagorean theorem, circle

The triangle is called rectangular if it has a right angle.
A right-angled triangle has two mutually perpendicular sides, called legs; its third party is called hypotenuse.

• According to the properties of the perpendicular and oblique, the hypotenuse is longer than each of the legs (but less than their sum).
• The sum of two acute angles of a right-angled triangle is equal to the right angle.
• Two heights of a right-angled triangle coincide with its legs. Therefore, one of the four remarkable points falls into the vertices of the right angle of the triangle.
• The center of the circumscribed circle of a right-angled triangle lies in the middle of the hypotenuse.
• The median of a right-angled triangle, drawn from the vertex of the right-angled angle to the hypotenuse, is the radius of the circle circumscribed about this triangle.

Consider an arbitrary right-angled triangle ABC and draw the height СD = hc from the vertex С of its right angle.

It will split this triangle into two right-angled triangles ACD and BCD; each of these triangles has a common acute angle with the triangle ABC and is therefore similar to the triangle ABC.

All three triangles ABC, ACD and BCD are similar to each other.

From the similarity of triangles, the following relations are determined:

• $$ h = \ sqrt (a_ (c) \ cdot b_ (c)) = \ frac (a \ cdot b) (c) $$;
• c = ac + bc;
• $$ a = \ sqrt (a_ (c) \ cdot c), b = \ sqrt (b_ (c) \ cdot c) $$;
• $$ (\ frac (a) (b)) ^ (2) = \ frac (a_ (c)) (b_ (c)) $$.

Pythagorean theorem one of the fundamental theorems of Euclidean geometry, establishing the relationship between the sides of a right-angled triangle.

Geometric formulation. In a right-angled triangle, the area of ​​the square built on the hypotenuse is equal to the sum of the areas of the squares built on the legs.

Algebraic formulation. In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the legs.
That is, denoting the length of the hypotenuse of the triangle through c, and the lengths of the legs through a and b:
a2 + b2 = c2

The converse theorem of Pythagoras.

Height of a right triangle

For any triple of positive numbers a, b and c such that
a2 + b2 = c2,
there is a right-angled triangle with legs a and b and a hypotenuse c.

Signs of equality of right-angled triangles:

• along the leg and hypotenuse;
• on two legs;
• along the leg and sharp corner;
• by hypotenuse and acute angle.

See also:
Area of ​​a triangle, isosceles triangle, equilateral triangle

Geometry. 8 Class. Test 4. Option 1 .

AD : CD = CD : BD. Hence CD2 = AD ∙ BD. They say:

AD : AC = AC : AB. Hence AC2 = AB ∙ AD. They say:

BD : BC = BC : AB. Hence BC2 = AB ∙ BD.

Solve the tasks:

1.

A) 70 cm; B) 55 cm; C) 65 cm; D) 45 cm; E) 53 cm.

2. The height of a right-angled triangle drawn to the hypotenuse divides the hypotenuse into segments 9 and 36.

Determine the length of this height.

A) 22,5; B) 19; C) 9; D) 12; E) 18.

4.

A) 30,25; B) 24,5; C) 18,45; D) 32; E) 32,25.

5.

A) 25; B) 24; C) 27; D) 26; E) 21.

6.

A) 8; B) 7; C) 6; D) 5; E) 4.

7.

8. The leg of a right-angled triangle is 30.

How do I find the height in a right triangle?

Find the distance from the vertex of a right angle to the hypotenuse if the radius of the circle circumscribed about this triangle is 17.

A) 17; B) 16; C) 15; D) 14; E) 12.

10.

A) 15; B) 18; C) 20; D) 16; E) 12.

A) 80; B) 72; C) 64; D) 81; E) 75.

12.

A) 7,5; B) 8; C) 6,25; D) 8,5; E) 7.

Compare answers!

D8.04.1. Proportional line segments in a right triangle

Geometry. 8 Class. Test 4. Option 1 .

В Δ АВС ∠АСВ = 90 °. AC and BC legs, AB hypotenuse.

CD is the height of the triangle drawn to the hypotenuse.

AD projection of the AC leg onto the hypotenuse,

BD projection of the BC leg onto the hypotenuse.

The height CD divides triangle ABC into two similar triangles (and to each other): Δ ADC and Δ CDB.

From the proportionality of the sides like Δ ADC and Δ CDB it follows:

AD : CD = CD : BD.

The property of the height of a right-angled triangle dropped by the hypotenuse.

Hence CD2 = AD ∙ BD. They say: the height of a right-angled triangle drawn to the hypotenuse,there is an average proportional value between the projections of the legs on the hypotenuse.

From the similarity between Δ ADC and Δ ACB it follows:

AD : AC = AC : AB. Hence AC2 = AB ∙ AD. They say: each leg is the average proportional value between the entire hypotenuse and the projection of this leg onto the hypotenuse.

Similarly, from the similarity of Δ СDВ and Δ АCB it follows:

BD : BC = BC : AB. Hence BC2 = AB ∙ BD.

Solve the tasks:

1. Find the height of a right-angled triangle drawn to the hypotenuse if it divides the hypotenuse into segments 25 cm and 81 cm.

A) 70 cm; B) 55 cm; C) 65 cm; D) 45 cm; E) 53 cm.

2. The height of a right-angled triangle, drawn to the hypotenuse, divides the hypotenuse into segments 9 and 36. Determine the length of this height.

A) 22,5; B) 19; C) 9; D) 12; E) 18.

4. The height of a right-angled triangle, drawn to the hypotenuse, is 22, the projection of one of the legs is 16. Find the projection of the other leg.

A) 30,25; B) 24,5; C) 18,45; D) 32; E) 32,25.

5. The leg of a right-angled triangle is 18, and its projection onto the hypotenuse is 12. Find the hypotenuse.

A) 25; B) 24; C) 27; D) 26; E) 21.

6. The hypotenuse is 32. Find the leg, the projection of which on the hypotenuse is 2.

A) 8; B) 7; C) 6; D) 5; E) 4.

7. The hypotenuse of a right-angled triangle is 45. Find the leg, the projection of which to the hypotenuse is 9.

8. The leg of a right-angled triangle is 30. Find the distance from the vertex of the right angle to the hypotenuse if the radius of the circle circumscribed about this triangle is 17.

A) 17; B) 16; C) 15; D) 14; E) 12.

10. The hypotenuse of a right-angled triangle is 41, and the projection of one of the legs is 16. Find the length of the height drawn from the vertex of the right angle to the hypotenuse.

A) 15; B) 18; C) 20; D) 16; E) 12.

A) 80; B) 72; C) 64; D) 81; E) 75.

12. The difference between the projections of the legs on the hypotenuse is 15, and the distance from the vertex of the right angle to the hypotenuse is 4. Find the radius of the circumscribed circle.

A) 7,5; B) 8; C) 6,25; D) 8,5; E) 7.

Triangles.

Basic concepts.

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

The segments are called parties, and points - peaks.

Sum of angles a triangle is equal to 180 º.

The height of the triangle.

Triangle height is a perpendicular drawn from the top to the opposite side.

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

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

In an obtuse triangle, the height is outside the triangle (Figure 3).

Triangle Height Properties:

The bisector of a triangle.

Bisector of a triangle is a line segment that divides the corner of the vertex in half and connects the vertex with a point on the opposite side (Fig. 5).

Bisector properties:

Median of the triangle.

Median of a triangle is a line 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 is the median drawn to the side a. In a right-angled triangle, the median drawn to the hypotenuse is half the hypotenuse: c m c = — 2 where m c- the median drawn to the hypotenuse c(Figure 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 2: 1 ratio, counting from the vertex. That is, the segment from the vertex to the center is twice as large as the segment from the center to the side of the triangle (Figure 9c). Three medians of a triangle divide it into six equal triangles.

The middle line of the triangle.

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

The middle line of the triangle is parallel to the third side and is 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 outer corner of a triangle is greater than any non-adjacent angle.

Right triangle.

Right triangle is a triangle with a right angle (fig. 12).

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

The other two parties are called legs.

Proportional line segments in a right-angled triangle.

1) In a right-angled triangle, the height drawn from a 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 is a triangle with two sides equal (Fig. 13).

These equal sides are called lateral sides and the third one is basis triangle.

V isosceles triangle base angles 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).

Equilateral triangle properties:

Wonderful properties of triangles.

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

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

Equality tests for 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 equal.

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

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

Triangle inequality.

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

Pythagorean theorem.

In a right-angled 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 by 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 by 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).

A triangle inscribed in a circle.

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

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

Sinus acute angle x opposing leg to the hypotenuse.
It is denoted like this: sinx.

Cosine acute angle x right triangle is the ratio adjacent leg to the hypotenuse.
It is denoted like this: cos x.

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

Cotangent acute angle x- This is the ratio of the adjacent leg to the opposite one.
It is denoted like this: ctgx.

Rules:

Leg opposite to the 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 to the 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 sharp angle x:

sin (90 ° - x) = cos x

cos (90 ° - x) = sin x

Right triangle- this is a triangle, one of the angles is straight, that is, it is equal to 90 degrees.

• The side opposite to the right angle is called the hypotenuse (in the figure it is indicated as c or AB)
• The side adjacent to the right angle is called the leg. Each right-angled triangle has two legs (indicated in the figure as a and b or AC and BC)

Right triangle formulas and properties

Formula designations:

(see picture above)

a, b- legs of a right triangle

c- hypotenuse

α, β - acute angles of the triangle

S- square

h- the height lowered from the top of the right angle to the hypotenuse

m a a from the opposite corner ( α )

m b is the median drawn to the side b from the opposite corner ( β )

m c is the median drawn to the side c from the opposite corner ( γ )

V right triangle any of the legs is less than the hypotenuse(Formulas 1 and 2). This property is a consequence of the Pythagorean theorem.

The cosine of any of the acute angles less than one (Formulas 3 and 4). This property follows from the previous one. Since any of the legs is less than the hypotenuse, then the ratio of the leg to the hypotenuse is always less than one.

The square of the hypotenuse is equal to the sum of the squares of the legs (Pythagorean theorem). (Formula 5). This property is constantly used when solving problems.

Area of ​​a right triangle equal to half the product of legs (Formula 6)

Sum of squares of medians to the legs, is equal to five squares of the median to the hypotenuse and five squares of the hypotenuse, divided by four (Formula 7). In addition to the above, there is 5 more formulas, therefore, it is recommended that you also familiarize yourself with the lesson "Median of a Right Triangle", which describes in more detail the properties of the median.

Height a right-angled triangle is equal to the product of the legs divided by the hypotenuse (Formula 8)

The squares of the legs are inversely proportional to the square of the height lowered to the hypotenuse (Formula 9). This identity is also one of the consequences of the Pythagorean theorem.

Hypotenuse length is equal to the diameter (two radii) of the circumscribed circle (Formula 10). Hypotenuse of a right triangle is the diameter of the circumscribed circle... This property is often used when solving problems.

Inscribed radius v right triangle circles can be found as half of an expression that includes the sum of the legs of this triangle minus the length of the hypotenuse. Or as the product of the legs, divided by the sum of all sides (perimeter) of a given triangle. (Formula 11)
Sine angle relation of the opposite this corner leg to hypotenuse(by definition of sine). (Formula 12). This property is used when solving problems. Knowing the size of the sides, you can find the angle that they form.

The cosine of the angle A (α, alpha) in a right-angled triangle will be equal to attitude adjacent this corner leg to hypotenuse(by definition of sine). (Formula 13)