Derivation of the formula for the height of a right triangle. Right triangle

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:

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, lowered to the hypotenuse, is equal to the geometric mean of the projections of the legs onto 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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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 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, so 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.

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)