How to find segments of the hypotenuse divided by height. Right triangle. Detailed theory with examples. Signs of equality of right triangles

Triangles.

Basic concepts.

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

The segments are called parties, and the points are peaks.

Sum of angles triangle is 180º.

Height of the triangle.

Triangle height- this is a perpendicular drawn from the vertex to the opposite side.

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

IN right triangle the legs are the altitudes of the triangle (Fig. 2).

In an obtuse triangle, the altitude extends outside the triangle (Fig. 3).

Properties of the altitude of a triangle:

Bisector of a triangle.

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

Properties of the bisector:

Median of a triangle.

Median of a triangle- 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 equal to half the hypotenuse:

c
m c = —
2

Where m c- median drawn to the hypotenuse c(Fig.9c)

The medians of the 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 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 (Fig. 9c).

The three medians of a triangle divide it into six equal triangles.

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 middle line of the triangle is parallel to the third side and equal to half of it

External angle of a triangle.

External corner triangle equal to the sum two non-adjacent internal corners (Fig. 11).

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

Right triangle.

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

Fig.14a

Isosceles triangle.

Isosceles triangle is a triangle whose two sides are equal (Fig. 13).

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

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

In an isosceles triangle, the median drawn to the base is both the bisector and the altitude 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 unique properties that will help you successfully solve problems involving these shapes. Some of these properties are outlined above. But we repeat them again, adding to them a few other wonderful features:

1) In a right triangle with angles of 90º, 30º and 60º legs b, lying opposite an angle of 30º, is equal to half of the hypotenuse. A lega more legb√3 times (Fig. 15 A). For example, if leg b is 5, then the hypotenuse c necessarily equals 10, and the leg A equals 5√3.

2) In a right isosceles triangle with angles of 90º, 45º and 45º, the hypotenuse is √2 times larger than 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 middle line parallel to it is 5.

4) In a right triangle, the median drawn to the hypotenuse is equal to half the hypotenuse (Fig. 9c): m c= s/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 intersection point of the medians is twice as large as the segment from the intersection point of the medians to the side of the triangle (Fig. 9c)

6) In a right triangle, the middle of the hypotenuse is the center of the circumscribed circle (Fig. 15 d).

Signs of equality of triangles.

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.

Second sign of equality: if a side and its adjacent angles of one triangle are equal to the side and its adjacent angles of another triangle, then such triangles are congruent.

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 altitude 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 · A.C. · 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 said to be 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 opposite leg to hypotenuse.
It is denoted as follows: sinx.

Cosine acute angle x of a right triangle is the ratio adjacent leg to hypotenuse.
Denoted as follows: cos x.

Tangent acute angle x- this is the ratio of the opposite side to the adjacent side.
It is designated as follows: tgx.

Cotangent acute angle x- this is the ratio of the adjacent side to the opposite side.
It is designated as follows: ctgx.

Rules:

Leg opposite the corner x, equal to the product hypotenuse on 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 the corner x, is equal to the product of the second leg by tg x:

b = a tg x

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

a = b· ctg x.

For any acute angle x:

sin (90° - x) = cos x

cos (90° - x) = sin x

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

Property: 2. The height of a right triangle dropped to the hypotenuse is equal to the average geometric projections legs to 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. A leg opposite 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 triangle, the median drawn to the hypotenuse is equal to half of it and equal to the radius of the circumscribed circle.

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

44. Theorem of cosines. Corollaries: relationship between diagonals and sides of a parallelogram; determining the type of triangle; formula for calculating the length of the median of a triangle; Calculation of the cosine of a triangle angle.

End of work -

This topic belongs to the section:

Class. Colloquium program on basic planimetry

Property of adjacent angles.. definition of two angles being adjacent if they have one side in common and 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 I really don’t want to, do I? 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, an opposite (for an angle) leg? Of course have! This is a leg!

What about the angle? Look carefully. Which leg is adjacent to the corner? Of course, the leg. This means that for the angle the leg is adjacent, and

Now, pay attention! Look what we got:

See how cool it is:

Now let's move on to tangent and cotangent.

How can I write this down in words now? What is the leg in relation to the angle? Opposite, of course - it “lies” opposite the corner. What about the leg? Adjacent to the corner. So what have we got?

See how the numerator and denominator have swapped places?

And now the corners again and made an exchange:

Summary

Let's briefly write down everything we've learned.

Pythagorean theorem:

The main theorem about right triangles is the Pythagorean theorem.

Pythagorean theorem

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

It is quite 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.

See how cleverly we divided its sides into lengths and!

Now let's connect the marked dots

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

What is the area of the larger square?

Right, .

What about 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 their hypotenuses.

What happened? Two rectangles. This means that the area of the “cuts” 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 side 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 side to the adjacent side.

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

And once again all this in the form of a tablet:

It is very comfortable!

Signs of equality of right triangles

I. On two sides

II. By leg and hypotenuse

III. By hypotenuse and acute angle

IV. Along the leg and acute angle

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

THEN 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 it was 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, three of their elements must be equal: two sides and the angle between them, two angles and the side between them, or three sides.

But for the equality of right triangles, only two corresponding elements are enough. Great, right?

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

Signs of similarity of right triangles

I. Along an acute angle

II. On two sides

III. By leg and hypotenuse

Median in a right triangle

Why is this so?

Instead of a right triangle, consider a whole rectangle.

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 turned out that

- median:

Remember this fact! Helps a lot!

What’s even more surprising is that the opposite is also true.

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

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

So let's start with this “besides...”.

Let's look at and.

But similar triangles have all equal angles!

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 us write down the relations of the corresponding parties:

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

Well, now, by 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:

You need to remember both of these formulas very well and use the one that is more convenient.

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 sides:
by 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 similarity of right triangles:

one acute corner: or
from the proportionality of 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 side 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 side to the adjacent side:
The cotangent of an acute angle of a right triangle is the ratio of the adjacent side to the opposite side: .

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:

via legs:

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

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

The sides that form a right angle are called legs. The picture shows the sides AD, DC and BD, DC- legs, and sides AC And NE- hypotenuse.

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

AB- hypotenuse;

AD And DВ

Triangle
There is a theorem:
comment system CACKLE

Solution: 1) The diagonals of any rectangle are equal. True 2) If a triangle has one acute angle, then this triangle is acute. Not true. Types of triangles. A triangle is called acute 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 formula for the height of a right triangle?

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 onto the hypotenuse (the segments into which the height divides the hypotenuse).

The altitude to the hypotenuse can be found through the area of a right triangle. If we apply the formula to find 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 equal to 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 by a and b, the length of the hypotenuse by c, the formula can be rewritten as

Since the radius of the circumcircle of a right triangle is equal to half the hypotenuse, the length of the altitude can be expressed in terms of the legs and the radius of the circumcircle:

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

From right triangle ABK

From right triangle ACK

The length of the altitude 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 its legs:

What is the formula for the height of a right triangle?

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 Unified State Exam?

The main theorem about right triangles is the Pythagorean theorem.

Pythagorean theorem

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

See how cleverly we divided its sides into lengths and!

Now let's connect the marked dots

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

What is the area of the larger square? Right, . What about 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 their hypotenuses. What happened? Two rectangles. This means that the area of the “cuts” 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 side 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 side to the adjacent side.

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

And once again all this in the form of a tablet:

Have you noticed one very convenient thing? Look at the sign 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! It is very important here that the legs are “appropriate”. For example, if it goes like this:

THEN 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 it was 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, three of their elements must be equal: two sides and the angle between them, two angles and the side between them, or three sides. But for the equality of right triangles, only two corresponding elements are enough. Great, right?

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

Signs of similarity of right triangles

III. By leg and hypotenuse

Median in a right triangle

Instead of a right triangle, consider a whole rectangle.

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

The intersection point of the diagonals is divided in half. The diagonals are equal.

And what follows from this?

So it turned out that

Remember this fact! Helps a lot!

What’s even more surprising is that the opposite is also true.

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

Let's start with this “besides.” "

But similar triangles have all equal angles!

The same can be said about and

Now let's draw it together:

They have the same sharp angles!

What benefit can be derived from this “triple” similarity?

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

Let us write down 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?

Now let's 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":

You need to remember both of these formulas very well and use the one that is more convenient. Let's write them down again

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

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Properties of a right triangle

Consider a right triangle (ABC) and its properties, which is presented in the figure. A right triangle has a hypotenuse - the side that lies opposite the right angle. The sides that form a right angle are called legs. The picture shows the sides AD, DC and BD, DC- legs, and sides AC And NE- 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 congruent.

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 acute angle of a right triangle are similar to the hypotenuse and acute angle of another triangle, then such triangles are congruent.

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 congruent.

Properties of a leg opposite an angle of 30°:

Theorem 1.

Height in a right triangle

In a right triangle with an angle of 30°, the leg opposite this angle will break half of the hypotenuse.

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

If the altitude 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 one and similar to one another. The following conclusions follow from this:

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

In a right triangle, the legs act as altitudes. The orthocenter is the point at which the intersection of the altitudes of the triangle occurs. It coincides with the vertex of the right angle of the figure.

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

AB- hypotenuse;

AD And DВ- segments that arise when dividing the hypotenuse by height.

Return to viewing information on the discipline "Geometry"

Triangle- This 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 its angles at 90° (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, right angle, leg, hypotenuse, Pythagorean theorem, circle

The triangle is called rectangular if it has a right angle.
A right triangle has two mutually perpendicular sides called legs; its third side 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 triangle is equal to a right angle.
Two altitudes of a right triangle coincide with its legs. Therefore, one of the four remarkable points falls at the vertices of the right angle of the triangle.
The circumcenter of a right triangle lies at the middle 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 the height CD = hc from the vertex C of its right angle.

It will split the given triangle into two right 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 formulation. 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 by c, and the lengths of the legs by a and b:
a2 + b2 = c2

Converse 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;
along the hypotenuse and acute angle.

See also:
Area of a triangle, 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 ∙ A.D. 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 altitude 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 the answers!

G8.04.1. 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 leg AC onto 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 it follows:

AD : CD = CD : B.D.

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

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

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

AD : AC = AC : AB. Hence AC2 = AB ∙ A.D. 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 altitude 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 altitude of a right triangle drawn to the hypotenuse divides the hypotenuse into segments 9 and 36. Determine the length of this altitude.

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

4. The altitude 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 to the hypotenuse is 12. Find the hypotenuse.

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

6. The hypotenuse is equal to 32. Find the side whose projection onto the hypotenuse is equal to 2.

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

7. The hypotenuse of a right triangle is 45. Find the side 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.

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

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

Formulas and properties of a right triangle

Formula designations:

(see picture above)

a, b- legs of a right triangle

c- hypotenuse

α, β - acute angles of a triangle

S- square

h- height lowered from the vertex of a right angle to the hypotenuse

m a a from the opposite corner ( α )

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

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

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

Cosine of any of the acute angles less than one (Formula 3 and 4). This property follows from the previous one. Since any of the legs is less than the hypotenuse, the ratio of leg to 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 squared 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 read the lesson “Median of a Right Triangle,” which describes the properties of the median in more detail.

Height of a right 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 equal to the diameter (two radii) of the circumscribed circle (Formula 10). Hypotenuse of a right triangle is the diameter of the circumcircle. This property is often used in problem solving.

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

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