Properties of the height lowered to the hypotenuse. Right triangle

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

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

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

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

Property: 6. Dependence between sides and angles of a right triangle:

44. Cosine theorem. Consequences: connection between diagonals and sides of a parallelogram; determining the type of triangle; formula for calculating the length of the median of a triangle; calculating the cosine of the angle of a triangle.

End of work -

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Class. Program of the Colloquium Fundamentals of Planimetry

The property of adjacent angles.. the definition of two angles are adjacent if one side they have in common in the other two form a straight line..

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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

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

- 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:

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(ABC) and its properties, which is shown in the figure. A right triangle has a hypotenuse, the side 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. Side drawing AD, DC and BD, DC- legs, and sides AC and SW- hypotenuse.

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

AB- hypotenuse;

AD and DB

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

Or, in another entry,

According to the Pythagorean theorem

What is the height in a right triangle formula

Height of a right triangle

The height of a right 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 on the hypotenuse (the segments into which the altitude divides the hypotenuse).

The altitude drawn to the hypotenuse can be found through the area of a right triangle. If we apply the formula for finding the area of a triangle

(half the product of a side and 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 twice the area of the triangle to the length of the hypotenuse:

Since the area of a right 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 triangle is equal to half 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 triangles, its length can be found through the ratios in the right triangle.

From right triangle ABK

From right triangle ACK

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

According to the Pythagorean theorem

If we square both sides of the equation:

You can get another formula for relating the height of a right triangle to 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 Examination or the OGE?

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

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.

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:

Have you noticed one very handy thing? Look at the plate carefully.

It is very comfortable!

Signs of equality of right triangles

II. By leg and hypotenuse

III. By hypotenuse and acute angle

IV. Along the leg and acute angle

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? Take a look at the “Triangle” 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

III. By leg and hypotenuse

Median in a right triangle

Consider a whole rectangle instead of a right triangle.

Draw a diagonal and consider the point where the diagonals intersect. What do you know about the diagonals of a rectangle?

Diagonal intersection point bisects Diagonals are equal

And what follows from this?

So it happened that

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

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

But in similar triangles all angles are equal!

The same can be said about and

Now let's draw it together:

Both have the same sharp corners!

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 The first formula "Height in a right triangle":

How to get a second one?

And now we apply the similarity of triangles and.

So, let's apply the similarity: .

What will happen now?

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

Both of these formulas must be remembered very well and the one that 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 triangle!

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Height property of a right triangle dropped to the hypotenuse

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Properties of a right triangle

Consider a right triangle (ABC) and its properties, which is shown in the figure. A right triangle has a hypotenuse, the side opposite the right angle. The sides that form a right angle are called legs. Side drawing AD, DC and BD, DC- legs, and sides AC and SW- hypotenuse.

Signs of equality of a right 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 congruent.

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 congruent.

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

Properties of a leg opposite an angle of 30 °:

Theorem 1.

Height in a right triangle

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

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

If the height is drawn from the vertex of the right angle to the hypotenuse, then such a triangle is divided into two smaller ones, similar to the outgoing and similar one to the other. The following conclusions follow from this:

The height is the geometric mean (mean proportional) of the two hypotenuse segments.
Each leg of the triangle is the mean proportional to the hypotenuse and adjacent segments.

In a right triangle, the legs act as heights. The orthocenter is the point where the heights of the triangle intersect. It coincides with the top of the right angle of the figure.

hC- the height coming out of the right angle of the triangle;

AB- hypotenuse;

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

Back to viewing references on the discipline "Geometry"

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

Triangle called rectangular if it has a right angle.
A right triangle has two mutually perpendicular sides called legs; the third side is called hypotenuse.

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

Consider an arbitrary right triangle ABC and draw a height CD = hc from the vertex C of its right angle.

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

Geometric wording. In a right 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 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 inverse Pythagorean theorem.

Height of a right triangle

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

Signs of equality of right triangles:

along the leg and hypotenuse;
on two legs;
along the leg and acute angle;
hypotenuse and acute angle.

See also:
Triangle Area, Isosceles Triangle, Equilateral Triangle

Geometry. 8 Class. Test 4. Option 1 .

AD : CD=CD : B.D. Hence CD2 = AD ∙ B.D. They say:

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

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

Solve problems:

A) 70 cm; b) 55 cm; c) 65 cm; D) 45 cm; e) 53 cm

2. The height of a right 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.

A) 30,25; b) 24,5; c) 18,45; D) 32; e) 32,25.

A) 25; b) 24; c) 27; D) 26; e) 21.

A) 8; b) 7; c) 6; D) 5; e) 4.

8. The leg of a right triangle is 30.

How to find the height in a right triangle?

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.

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.

Check answers!

D8.04.1. Proportional segments in a right triangle

Geometry. 8 Class. Test 4. Option 1 .

In Δ ABC ∠ACV = 90°. AC and BC legs, AB hypotenuse.

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

AD projection of the AC leg on the hypotenuse,

BD projection of the BC leg onto the hypotenuse.

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

From the proportionality of the sides of similar Δ ADC and Δ CDB follows:

AD : CD=CD : B.D.

Property of the height of a right triangle dropped to the hypotenuse.

Hence CD2 = AD ∙ B.D. They say: the height of a right triangle drawn to the hypotenuse,is the average proportional value between the projections of the legs on the hypotenuse.

From the similarity of 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 CDB and Δ ACB it follows:

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

Solve problems:

1. Find the height of a right 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 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 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 triangle is 18, and its projection on 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 whose projection onto the hypotenuse is 2.

A) 8; b) 7; c) 6; D) 5; e) 4.

7. The hypotenuse of a right triangle is 45. Find the leg whose projection onto the hypotenuse is 9.

8. The leg of a right 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 triangle is 41, and the projection of one of the legs is 16. Find the length of the altitude 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 in the projections of the legs onto 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.