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 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)
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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 there is! 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:
Resume
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's very convenient!
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
a) 
b) 
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.
It is necessary that 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?
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:
When solving geometric problems, it is useful to follow such an algorithm. While reading the conditions of the problem, it is necessary
- Make a drawing. The drawing should correspond as much as possible to the conditions of the problem, so its main task is to help find the solution
- Put all the data from the problem statement on the drawing
- Write down all the geometric concepts that appear in the problem
- Remember all the theorems that relate to these concepts
- Draw on the drawing all the relationships between the elements of a geometric figure that follow from these theorems
For example, if the problem contains the words bisector of an angle of a triangle, you need to remember the definition and properties of a bisector and indicate equal or proportional segments and angles in the drawing.
In this article you will find the basic properties of a triangle that you need to know to successfully solve problems.
TRIANGLE.
Area of a triangle.
1. ,
here - an arbitrary side of the triangle, - the height lowered to this side.
2.
,
here and are arbitrary sides of the triangle, and is the angle between these sides:
3. Heron's formula:
Here are the lengths of the sides of the triangle, is the semi-perimeter of the triangle,
4. ,
here is the semi-perimeter of the triangle, and is the radius of the inscribed circle.
Let be the lengths of the tangent segments.
Then Heron's formula can be written as follows:
6. ,
here - the lengths of the sides of the triangle, - the radius of the circumscribed circle.
If a point is taken on the side of a triangle that divides this side in the ratio m: n, then the segment connecting this point with the vertex of the opposite angle divides the triangle into two triangles, the areas of which are in the ratio m: n:
The ratio of the areas of similar triangles is equal to the square of the similarity coefficient.
Median of a triangle
This is a segment connecting the vertex of a triangle to the middle of the opposite side.
Medians of a triangle intersect at one point and are divided by the intersection point in a ratio of 2:1, counting from the vertex.
The intersection point of the medians of a regular triangle divides the median into two segments, the smaller of which is equal to the radius of the inscribed circle, and the larger of which is equal to the radius of the circumscribed circle.
The radius of the circumscribed circle is twice the radius of the inscribed circle: R=2r
Median length arbitrary triangle
,
here - the median drawn to the side - the lengths of the sides of the triangle.
Bisector of a triangle
This is the bisector segment of any angle of a triangle connecting the vertex of this angle with the opposite side.
Bisector of a triangle divides a side into segments proportional to the adjacent sides:
Bisectors of a triangle intersect at one point, which is the center of the inscribed circle.
All points of the angle bisector are equidistant from the sides of the angle.
Triangle height
This is a perpendicular segment dropped from the vertex of the triangle to the opposite side, or its continuation. In an obtuse triangle, the altitude drawn from the vertex of the acute angle lies outside the triangle.
The altitudes of a triangle intersect at one point, which is called orthocenter of the triangle.
To find the height of a triangle drawn to the side, you need to find its area in any available way, and then use the formula:
Center of the circumcircle of a triangle, lies at the point of intersection of the perpendicular bisectors drawn to the sides of the triangle.
Circumference radius of a triangle can be found using the following formulas:
Here are the lengths of the sides of the triangle, and is the area of the triangle.
,
where is the length of the side of the triangle and is the opposite angle. (This formula follows from the sine theorem.)
Triangle inequality
Each side of the triangle is less than the sum and greater than the difference of the other two.
The sum of the lengths of any two sides is always greater than the length of the third side:
Opposite the larger side lies the larger angle; Opposite the larger angle lies the larger side:
If , then vice versa.
Theorem of sines:
The sides of a triangle are proportional to the sines of the opposite angles:
Cosine theorem:
The square of a side of a triangle is equal to the sum of the squares of the other two sides without twice the product of these sides by the cosine of the angle between them:
Right triangle
- This is a triangle, one of the angles of which is 90°.
The sum of the acute angles of a right triangle is 90°.
The hypotenuse is the side that lies opposite the 90° angle. The hypotenuse is the longest side.
Pythagorean theorem:
the square of the hypotenuse is equal to the sum of the squares of the legs: 
The radius of a circle inscribed in a right triangle is equal to
,
here is the radius of the inscribed circle, - the legs, - the hypotenuse:
Center of the circumcircle of a right triangle lies in the middle of the hypotenuse:
Median of a right triangle drawn to the hypotenuse, is equal to half the hypotenuse.
Definition of sine, cosine, tangent and cotangent of a right triangle look
The ratio of elements in a right triangle:
The square of the altitude of a right triangle drawn from the vertex of a right angle is equal to the product of the projections of the legs onto the hypotenuse:
The square of the leg is equal to the product of the hypotenuse and the projection of the leg onto the hypotenuse:
Leg lying opposite the corner equal to half the hypotenuse:
Isosceles triangle.
The bisector of an isosceles triangle drawn to the base is the median and altitude.
In an isosceles triangle, the base angles are equal.
Apex angle.
And - sides,
And - angles at the base.
Height, bisector and median.
Attention! The height, bisector and median drawn to the side do not coincide.
Regular triangle
(or equilateral triangle ) is a triangle, all sides and angles of which are equal to each other.
Area of a regular triangle equal to
where is the length of the side of the triangle.
Center of a circle inscribed in a regular triangle, coincides with the center of the circle circumscribed about a regular triangle and lies at the point of intersection of the medians.
Intersection point of the medians of a regular triangle divides the median into two segments, the smaller of which is equal to the radius of the inscribed circle, and the larger of which is equal to the radius of the circumscribed circle.
If one of the angles of an isosceles triangle is 60°, then the triangle is regular.
Middle line of the triangle
This is a segment connecting the midpoints of two sides.
In the figure DE is the middle line of triangle ABC.
The middle line of the triangle is parallel to the third side and equal to its half: DE||AC, AC=2DE
External angle of a triangle
This is the angle adjacent to any angle of the triangle.
An exterior angle of a triangle is equal to the sum of two angles not adjacent to it.
External angle trigonometric functions:
Signs of equality of triangles:
1 . If two sides and the angle between them of one triangle are respectively equal to two sides and the angle between them of another triangle, then such triangles are congruent.
2 . If a side and two adjacent angles of one triangle are respectively equal to a side and two adjacent angles of another triangle, then such triangles are congruent.
3 If three sides of one triangle are respectively equal to three sides of another triangle, then such triangles are congruent.
Important: since in a right triangle two angles are obviously equal, then for equality of two right triangles equality of only two elements is required: two sides, or a side and an acute angle.
Signs of similarity of triangles:
1 . If two sides of one triangle are proportional to two sides of another triangle, and the angles between these sides are equal, then these triangles are similar.
2 . If three sides of one triangle are proportional to three sides of another triangle, then the triangles are similar.
3 . If two angles of one triangle are equal to two angles of another triangle, then the triangles are similar.
Important: In similar triangles, similar sides lie opposite equal angles.
Menelaus' theorem
Let a line intersect a triangle, and is the point of its intersection with side , is the point of its intersection with side , and is the point of its intersection with the continuation of side . Then
