228. In this chapter, we will mainly understand by the notation of the segments AB, AC, etc., the numbers expressing them.
We know (n. 226) that if two segments a and b are given geometrically, then we can construct an average proportional between them. Now let the segments be given not geometrically, but by numbers, i.e., by a and b we will understand numbers expressing 2 given segments. Then finding the average proportional segment will be reduced to finding the number x from the proportion a/x = x/b, where a, b and x are numbers. From this proportion we have:
x 2 = ab
x = √ab
229. Let's have a right-angled triangle ABC (drawing 224).
Let us drop the perpendicular BD from the vertex of its right angle (∠B right angle) to the hypotenuse AC. Then from item 225 we know:
1) AC/AB = AB/AD and 2) AC/BC = BC/DC.
From here we get:
AB 2 = AC AD and BC 2 = AC DC.
Adding the equalities obtained in parts, we get:
AB 2 + BC 2 \u003d AC AD + AC DC \u003d AC (AD + DC).
i.e. the square of the number expressing the hypotenuse is equal to the sum of the squares of the numbers expressing the legs of a right triangle.
In short they say: The square of the hypotenuse of a right triangle is equal to the sum of the squares of the legs.
If we give the resulting formula a geometric interpretation, then we get the Pythagorean theorem already known to us (section 161):
the square built on the hypotenuse of a right triangle is equal to the sum of the squares built on the legs.
From the equation AB 2 + BC 2 \u003d AC 2, sometimes you have to find the leg of a right triangle, along the hypotenuse and the other leg. We get, for example:
AB 2 \u003d AC 2 - BC 2 and, consequently,
230. The found numerical ratio between the sides of a right-angled triangle allows solving many computational problems. Let's solve some of them:
1. Calculate the area of an equilateral triangle given its side.
Let ∆ABC (Ch. 225) be equilateral and each of its sides be expressed by the number a (AB = BC = AC = a). To calculate the area of this triangle, you must first find out its height BD, which we will call through h. We know that in an equilateral triangle the height BD bisects the base AC, i.e. AD = DC = a/2. Therefore, from a right triangle DBC we have:
BD 2 \u003d BC 2 - DC 2,
h 2 \u003d a 2 - a 2 / 4 \u003d 3a 2 / 4 (we perform the subtraction).
Hence we have:
(we take out the multiplier from under the root).
Therefore, calling the number expressing the area of our triangle through Q and knowing that the area is ∆ABC = (AC BD)/2, we find:
We can look at this formula as one of the ways to measure the area of an equilateral triangle: we need to measure its side in linear units, square the found number, multiply the resulting number by √3 and divide by 4 - we get the expression for the area in square (corresponding) units.
2. The sides of the triangle are 10, 17 and 21 lines. single Calculate its area.
Let us lower the height h in our triangle (Ch. 226) to the larger side - it will certainly pass inside the triangle, since in a triangle an obtuse angle can only be located opposite the larger side. Then the larger side, = 21, will be divided into 2 segments, one of which will be denoted by x (see drawing) - then the other = 21 - x. We get two right triangles, of which we have:
h 2 \u003d 10 2 - x 2 and h 2 \u003d 17 2 - (21 - x) 2
Since the left sides of these equations are the same, then
10 2 - x 2 \u003d 17 2 - (21 - x) 2
By doing the following we get:
10 2 - x 2 \u003d 289 - 441 + 42x - x 2
Simplifying this equation, we find:
Then from the equation h 2 \u003d 10 2 - x 2, we get:
h 2 \u003d 10 2 - 6 2 \u003d 64
and hence
Then the required area is found:
Q = (21 8)/2 quad. single = 84 sq. single
3. You can solve the general problem:
How to calculate the area of a triangle given its sides?
Let the sides of the triangle ABC be expressed by the numbers BC = a, AC = b and AB = c (Chart 227). Let us assume that AC is the big side; then the height BD will go inside ∆ABC. Let's call: BD = h, DC = x and then AD = b - x.
From ∆BDC we have: h 2 = a 2 – x 2 .
From ∆ABD we have: h 2 = c 2 - (b - x) 2 ,
whence a 2 - x 2 \u003d c 2 - (b - x) 2.
Solving this equation, we successively obtain:
2bx \u003d a 2 + b 2 - c 2 and x \u003d (a 2 + b 2 - c 2) / 2b.
(The latter is written on the basis that the numerator 4a 2 b 2 - (a 2 + b 2 - c 2) 2 can be considered as an equality of squares, which we decompose into the product of the sum and the difference).
This formula is transformed by introducing the perimeter of the triangle, which we denote by 2p, i.e.
Subtracting 2c from both sides of the equation, we get:
a + b + c - 2c = 2p - 2c or a + b - c = 2(p - c):
We will also find:
c + a - b = 2(p - b) and c - a + b = 2(p - a).
Then we get:
(p expresses the half-perimeter of the triangle).
This formula can be used to calculate the area of a triangle given its three sides.
231. Exercises.
232. In § 229 we found the relationship between the sides of a right triangle. You can find a similar dependence for the sides (with the addition of another segment) of an oblique triangle.
Let us first have ∆ABC (Ch. 228) such that ∠A is sharp. Let us try to find an expression for the square of the side BC lying opposite this acute angle (similarly to how we found the expression for the square of the hypotenuse in § 229).
Constructing BD ⊥ AC, we obtain from the right triangle BDC:
BC 2 = BD 2 + DC 2
Let's replace BD2 by defining it from ABD, whence we have:
BD 2 \u003d AB 2 - AD 2,
and the segment DC is replaced by AC - AD (obviously, DC = AC - AD). Then we get:
BC 2 = AB 2 - AD 2 + (AC - AD) 2 = AB 2 - AD 2 + AC 2 - 2AC AD + AD 2
Having carried out the reduction of similar terms, we find:
BC 2 \u003d AB 2 + AC 2 - 2AC AD.
This formula reads: the square of the side of a triangle opposite an acute angle is equal to the sum of the squares of its other two sides, minus twice the product of one of these sides and its segment from the vertex of the acute angle to the height.
233. Let now ∠A and ∆ABC (Ch. 229) be obtuse. Let us find an expression for the square of the side BC lying opposite the obtuse angle.
Having built the height BD, it will now be located somewhat differently: at 228 where ∠A is sharp, points D and C are located on the same side of A, and here, where ∠A is obtuse, points D and C will be located on opposite sides of A. Then from a rectangular ∆BDC we get:
BC 2 = BD 2 + DC 2
We can replace BD2 by defining it from the rectangular ∆BDA:
BD 2 \u003d AB 2 - AD 2,
and the segment DC = AC + AD, which is obvious. Replacing, we get:
BC 2 = AB 2 - AD 2 + (AC + AD) 2 = AB 2 - AD 2 + AC 2 + 2AC AD + AD 2
Performing the reduction of similar terms, we find:
BC 2 = AB 2 + AC 2 + 2AC AD,
i.e. the square of the side of a triangle opposite an obtuse angle is equal to the sum of the squares of its other two sides, plus twice the product of one of them and its segment from the vertex of the obtuse angle to the height.
This formula, as well as the formula of item 232, admits a geometric interpretation, which is easy to find.
234. Using the properties of paragraphs. 229, 232, 233, we can, if we are given the sides of a triangle in numbers, find out if this triangle has a right or obtuse angle.
A right or obtuse angle in a triangle can only be located opposite the larger side, what is the angle opposite it, it is easy to find out: this angle is acute, right or obtuse, depending on whether the square of the larger side is less than, equal to or greater than the sum of the squares of the other two sides .
Find out if there is a right or obtuse angle in the following triangles, defined by their sides:
1) 15 dm., 13 dm. and 14 dm.; 2) 20, 29 and 21; 3) 11, 8 and 13; 4) 7, 11 and 15.
235. Let's have a parallelogram ABCD (drawing 230); construct its diagonals AC and BD and its heights BK ⊥ AD and CL ⊥ AD.
Then if ∠A (∠BAD) is acute, then ∠D (∠ADC) is necessarily obtuse (because their sum = 2d). From ∆ABD, where ∠A is considered sharp, we have:
BD 2 \u003d AB 2 + AD 2 - 2AD AK,
and from ∆ACD, where ∠D is obtuse, we have:
AC 2 = AD 2 + CD 2 + 2AD DL.
Let us replace the segment AD in the last formula with the segment BC equal to it and DL equal to it AK (DL = AK, since ∆ABK = ∆DCL, which is easy to see). Then we get:
AC2 = BC2 + CD2 + 2AD AK.
Adding the expression for BD2 with the last expression for AC 2 , we find:
BD 2 + AC 2 \u003d AB 2 + AD 2 + BC 2 + CD 2,
since the terms –2AD AK and +2AD AK cancel each other out. The resulting equality can be read:
The sum of the squares of the diagonals of a parallelogram is equal to the sum of the squares of its sides.
236. Calculating the median and bisector of a triangle along its sides. Let the median BM (that is, AM = MC) be constructed in the triangle ABC (Ch. 231). Knowing the sides ∆ABC: BC = a, AC = b and AB = c, calculate the median BM.
We continue BM and postpone the segment MD = BM. Connecting D to A and D to C, we get the parallelogram ABCD (this is easy to figure out, since ∆AMD = ∆BMC and ∆AMB = ∆DMC).
Calling the median BM through m, we get BD = 2m and then, using the previous paragraph, we have:
237. Calculation of the radius circumscribed about the triangle of a circle. Let a circle O be described near ∆ABC (Ch. 233). Let us construct the diameter of the circle BD, the chord AD and the height of the triangle BH.
Then ∆ABD ~ ∆BCH (∠A = ∠H = d - angle A is right, because it is inscribed, based on diameter BD and ∠D = ∠C, as inscribed, based on one arc AB). Therefore we have:
or, calling the radius OB by R, the height BH by h, and the sides AB and BC, as before, by c and a respectively:
but the area is ∆ABC = Q = bh/2, whence h = 2Q/b.
Therefore, R = (abc) / (4Q).
We are able (item 230 task 3) to calculate the area of a triangle Q on its sides. From here we can calculate R for the three sides of the triangle.
238. Calculation of the radius of a circle inscribed in a triangle. Let us inscribe in ∆ABC, the sides of which are given (Ch. 234), the circle O. Connecting its center O with the vertices of the triangle and with the points of contact D, E and F of the sides to the circle, we find that the radii of the circle OD, OE and OF serve as the heights of the triangles BOC, COA and AOB.
Calling the radius of the inscribed circle through r, we have:
Two triangles are said to be congruent if they can be overlapped. Figure 1 shows equal triangles ABC and A 1 B 1 C 1. Each of these triangles can be superimposed on another so that they are completely compatible, that is, their vertices and sides are paired together. It is clear that in this case the angles of these triangles will be combined in pairs.
Thus, if two triangles are equal, then the elements (i.e., sides and angles) of one triangle are respectively equal to the elements of the other triangle. Note that in equal triangles against respectively equal sides(i.e. overlapping when superimposed) lie equal angles and back: opposite correspondingly equal angles lie equal sides.
So, for example, in equal triangles ABC and A 1 B 1 C 1, shown in Figure 1, equal angles C and C 1 lie against respectively equal sides AB and A 1 B 1. The equality of triangles ABC and A 1 B 1 C 1 will be denoted as follows: Δ ABC = Δ A 1 B 1 C 1. It turns out that the equality of two triangles can be established by comparing some of their elements.
Theorem 1. The first sign of equality of triangles. 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 equal (Fig. 2).
Proof. Consider triangles ABC and A 1 B 1 C 1, in which AB \u003d A 1 B 1, AC \u003d A 1 C 1 ∠ A \u003d ∠ A 1 (see Fig. 2). Let us prove that Δ ABC = Δ A 1 B 1 C 1 .
Since ∠ A \u003d ∠ A 1, then the triangle ABC can be superimposed on the triangle A 1 B 1 C 1 so that the vertex A is aligned with the vertex A 1, and the sides AB and AC overlap, respectively, on the rays A 1 B 1 and A 1 C one . Since AB \u003d A 1 B 1, AC \u003d A 1 C 1, then side AB will be combined with side A 1 B 1 and side AC - with side A 1 C 1; in particular, points B and B 1 , C and C 1 will coincide. Therefore, the sides BC and B 1 C 1 will be aligned. So, triangles ABC and A 1 B 1 C 1 are completely compatible, which means they are equal.
Theorem 2 is proved similarly by the superposition method.
Theorem 2. The second sign of the equality of triangles. If the side and two angles adjacent to it of one triangle are respectively equal to the side and two angles adjacent to it of another triangle, then such triangles are equal (Fig. 34).
Comment. Based on Theorem 2, Theorem 3 is established.
Theorem 3. The sum of any two interior angles of a triangle is less than 180°.
Theorem 4 follows from the last theorem.
Theorem 4. An external angle of a triangle is greater than any internal angle not adjacent to it.
Theorem 5. The third sign of the equality of triangles. If three sides of one triangle are respectively equal to three sides of another triangle, then such triangles are equal ().
Example 1 In triangles ABC and DEF (Fig. 4)
∠ A = ∠ E, AB = 20 cm, AC = 18 cm, DE = 18 cm, EF = 20 cm. Compare triangles ABC and DEF. What angle in triangle DEF is equal to angle B?
Decision. These triangles are equal in the first sign. Angle F of triangle DEF is equal to angle B of triangle ABC, since these angles lie opposite the corresponding equal sides DE and AC.
Example 2 Segments AB and CD (Fig. 5) intersect at point O, which is the midpoint of each of them. What is segment BD equal to if segment AC is 6 m?
Decision.
Triangles AOC and BOD are equal (by the first criterion): ∠ AOC = ∠ BOD (vertical), AO = OB, CO = OD (by condition).
From the equality of these triangles follows the equality of their sides, i.e. AC = BD. But since, according to the condition, AC = 6 m, then BD = 6 m.
The science of geometry tells us what a triangle, square, cube is. In the modern world, it is studied in schools by everyone without exception. Also, a science that directly studies what a triangle is and what properties it has is trigonometry. She explores in detail all the phenomena associated with data. We will talk about what a triangle is today in our article. Their types will be described below, as well as some theorems related to them.
What is a triangle? Definition
This is a flat polygon. It has three corners, which is clear from its name. It also has three sides and three vertices, the first of which are segments, the second are points. Knowing what two angles are equal to, you can find the third one by subtracting the sum of the first two from the number 180.
What are triangles?
They can be classified according to various criteria.
First of all, they are divided into acute-angled, obtuse-angled and rectangular. The first have acute angles, that is, those that are less than 90 degrees. In obtuse angles, one of the angles is obtuse, that is, one that is equal to more than 90 degrees, the other two are acute. Acute triangles also include equilateral triangles. Such triangles have all sides and angles equal. They are all equal to 60 degrees, this can be easily calculated by dividing the sum of all angles (180) by three.
Right triangle
It is impossible not to talk about what a right triangle is.
Such a figure has one angle equal to 90 degrees (straight), that is, two of its sides are perpendicular. The other two angles are acute. They can be equal, then it will be isosceles. The Pythagorean theorem is related to the right triangle. With its help, you can find the third side, knowing the first two. According to this theorem, if you add the square of one leg to the square of the other, you can get the square of the hypotenuse. The square of the leg can be calculated by subtracting the square of the known leg from the square of the hypotenuse. Speaking about what a triangle is, we can recall the isosceles. This is one in which two of the sides are equal, and two of the angles are also equal.
What is the leg and hypotenuse?
The leg is one of the sides of a triangle that form an angle of 90 degrees. The hypotenuse is the remaining side that is opposite the right angle. From it, a perpendicular can be lowered onto the leg. The ratio of the adjacent leg to the hypotenuse is called the cosine, and the opposite is called the sine.
- what are its features?
It is rectangular. Its legs are three and four, and the hypotenuse is five. If you saw that the legs of this triangle are equal to three and four, you can be sure that the hypotenuse will be equal to five. Also, according to this principle, it can be easily determined that the leg will be equal to three if the second is equal to four, and the hypotenuse is five. To prove this statement, you can apply the Pythagorean theorem. If two legs are 3 and 4, then 9 + 16 \u003d 25, the root of 25 is 5, that is, the hypotenuse is 5. Also, the Egyptian triangle is called a right triangle, whose sides are 6, 8 and 10; 9, 12 and 15 and other numbers with a ratio of 3:4:5.
What else could be a triangle?
Triangles can also be inscribed and circumscribed. The figure around which the circle is described is called inscribed, all its vertices are points lying on the circle. A circumscribed triangle is one in which a circle is inscribed. All its sides are in contact with it at certain points.
How is
The area of any figure is measured in square units (square meters, square millimeters, square centimeters, square decimeters, etc.). This value can be calculated in a variety of ways, depending on the type of triangle. The area of any figure with angles can be found by multiplying its side by the perpendicular dropped onto it from the opposite angle, and dividing this figure by two. You can also find this value by multiplying the two sides. Then multiply this number by the sine of the angle between these sides, and divide this by two. Knowing all the sides of a triangle, but not knowing its angles, you can find the area in another way. To do this, you need to find half the perimeter. Then alternately subtract different sides from this number and multiply the four values obtained. Next, find out the number that came out. The area of an inscribed triangle can be found by multiplying all the sides and dividing the resulting number by which is circumscribed around it times four.
The area of the described triangle is found in this way: we multiply half the perimeter by the radius of the circle that is inscribed in it. If then its area can be found as follows: we square the side, multiply the resulting figure by the root of three, then divide this number by four. Similarly, you can calculate the height of a triangle in which all sides are equal, for this you need to multiply one of them by the root of three, and then divide this number by two.
Triangle theorems
The main theorems that are associated with this figure are the Pythagorean theorem, described above, and cosines. The second (sine) is that if you divide any side by the sine of the angle opposite to it, you can get the radius of the circle that is described around it, multiplied by two. The third (cosine) is that if the sum of the squares of the two sides is subtracted from their product, multiplied by two and the cosine of the angle located between them, then the square of the third side will be obtained.
Dali triangle - what is it?
Many, faced with this concept, at first think that this is some kind of definition in geometry, but this is not at all the case. The Dali Triangle is the common name for three places that are closely associated with the life of the famous artist. Its “tops” are the house where Salvador Dali lived, the castle that he gave to his wife, and the museum of surrealistic paintings. During a tour of these places, you can learn many interesting facts about this original creative artist, known throughout the world.
The simplest polygon that is studied at school is a triangle. It is more understandable for students and encounters fewer difficulties. Despite the fact that there are different types of triangles that have special properties.
What shape is called a triangle?
Formed by three points and line segments. The former are called vertices, the latter are called sides. Moreover, all three segments must be connected so that corners form between them. Hence the name of the figure "triangle".
Differences in the names in the corners
Since they can be sharp, obtuse and straight, the types of triangles are determined by these names. Accordingly, there are three groups of such figures.
- First. If all the angles of a triangle are acute, then it will be called an acute triangle. Everything is logical.
- Second. One of the angles is obtuse, so the triangle is obtuse. Easier nowhere.
- Third. There is an angle equal to 90 degrees, which is called a right angle. The triangle becomes rectangular.
Differences in names on the sides
Depending on the features of the sides, the following types of triangles are distinguished:
the general case is versatile, in which all sides have an arbitrary length;
isosceles, two sides of which have the same numerical values;
equilateral, the lengths of all its sides are the same.
If the task does not specify a specific type of triangle, then you need to draw an arbitrary one. In which all angles are acute, and the sides have different lengths.
Properties common to all triangles
- If you add up all the angles of a triangle, you get a number equal to 180º. And it doesn't matter what kind it is. This rule always applies.
- The numerical value of any side of the triangle is less than the other two added together. Moreover, it is greater than their difference.
- Each outer corner has a value that is obtained by adding two inner corners that are not adjacent to it. Moreover, it is always larger than the adjacent internal one.
- The smallest side of a triangle is always opposite the smallest angle. Conversely, if the side is large, then the angle will be the largest.
These properties are always valid, no matter what types of triangles are considered in problems. All the rest follow from specific features.
Properties of an isosceles triangle
- The angles adjacent to the base are equal.
- The height that is drawn to the base is also the median and the bisector.
- The heights, medians and bisectors, which are built to the sides of the triangle, are respectively equal to each other.
Properties of an equilateral triangle
If there is such a figure, then all the properties described a little above will be true. Because an equilateral will always be an isosceles one. But not vice versa, an isosceles triangle will not necessarily be equilateral.
- All its angles are equal to each other and have a value of 60º.
- Any median of an equilateral triangle is its height and bisector. And they are all equal to each other. To determine their values, there is a formula that consists of the product of the side and the square root of 3 divided by 2.
Properties of a right triangle
- Two acute angles add up to 90º.
- The length of the hypotenuse is always greater than that of any of the legs.
- The numerical value of the median drawn to the hypotenuse is equal to half of it.
- The leg is equal to the same value if it lies opposite an angle of 30º.
- The height, which is drawn from the top with a value of 90º, has a certain mathematical dependence on the legs: 1 / n 2 \u003d 1 / a 2 + 1 / in 2. Here: a, c - legs, n - height.
Problems with different types of triangles
No. 1. Given an isosceles triangle. Its perimeter is known and is equal to 90 cm. It is required to know its sides. As an additional condition: the lateral side is 1.2 times smaller than the base.
The value of the perimeter directly depends on the quantities that need to be found. The sum of all three sides will give 90 cm. Now you need to remember the sign of a triangle, according to which it is isosceles. That is, the two sides are equal. You can make an equation with two unknowns: 2a + b \u003d 90. Here a is the side, b is the base.
It's time for an additional condition. Following it, the second equation is obtained: b \u003d 1.2a. You can substitute this expression into the first one. It turns out: 2a + 1.2a \u003d 90. After transformations: 3.2a \u003d 90. Hence a \u003d 28.125 (cm). Now it's easy to find out the reason. It is best to do this from the second condition: v \u003d 1.2 * 28.125 \u003d 33.75 (cm).
To check, you can add three values: 28.125 * 2 + 33.75 = 90 (cm). All right.
Answer: the sides of the triangle are 28.125 cm, 28.125 cm, 33.75 cm.
No. 2. The side of an equilateral triangle is 12 cm. You need to calculate its height.
Decision. To search for an answer, it is enough to return to the moment where the properties of the triangle were described. This is the formula for finding the height, median and bisector of an equilateral triangle.
n \u003d a * √3 / 2, where n is the height, a is the side.
Substitution and calculation give the following result: n = 6 √3 (cm).
This formula does not need to be memorized. Suffice it to recall that the height divides the triangle into two rectangular ones. Moreover, it turns out to be a leg, and the hypotenuse in it is the side of the original one, the second leg is half of the known side. Now you need to write down the Pythagorean theorem and derive a formula for the height.
Answer: the height is 6 √3 cm.
No. 3. MKR is given - a triangle, 90 degrees in which makes an angle K. The sides MP and KR are known, they are equal to 30 and 15 cm, respectively. You need to find out the value of the angle P.
Decision. If you make a drawing, it becomes clear that MP is the hypotenuse. Moreover, it is twice as large as the leg of the CD. Again, you need to turn to the properties. One of them is just related to the corners. From it it is clear that the angle of the KMR is 30º. So the desired angle P will be equal to 60º. This follows from another property which states that the sum of two acute angles must equal 90º.
Answer: angle R is 60º.
No. 4. You need to find all the angles of an isosceles triangle. It is known about him that the external angle from the angle at the base is 110º.
Decision. Since only the outer corner is given, this should be used. It forms with an internal angle developed. So they add up to 180º. That is, the angle at the base of the triangle will be equal to 70º. Since it is isosceles, the second angle has the same value. It remains to calculate the third angle. By a property common to all triangles, the sum of the angles is 180º. So the third is defined as 180º - 70º - 70º = 40º.
Answer: the angles are 70º, 70º, 40º.
No. 5. It is known that in an isosceles triangle the angle opposite the base is 90º. A dot is marked on the base. The segment connecting it with a right angle divides it in a ratio of 1 to 4. You need to know all the angles of the smaller triangle.
Decision. One of the corners can be determined immediately. Since the triangle is right-angled and isosceles, those that lie at its base will be 45º, that is, 90º / 2.
The second of them will help to find the relation known in the condition. Since it is equal to 1 to 4, then there are only 5 parts into which it is divided. So, to find out the smaller angle of the triangle, you need 90º / 5 = 18º. It remains to find out the third. To do this, from 180º (the sum of all the angles of a triangle), you need to subtract 45º and 18º. The calculations are simple, and it turns out: 117º.
Triangle - definition and general concepts
A triangle is such a simple polygon, consisting of three sides and having the same number of angles. Its planes are limited by 3 points and 3 segments connecting these points in pairs.
All vertices of any triangle, regardless of its variety, are indicated by capital Latin letters, and its sides are depicted by the corresponding designations of opposite vertices, only not in capital letters, but in small letters. So, for example, a triangle with vertices labeled A, B, and C has sides a, b, c.
If we consider a triangle in Euclidean space, then this is such a geometric figure that was formed using three segments connecting three points that do not lie on one straight line.
Look closely at the picture above. On it, points A, B and C are the vertices of this triangle, and its segments are called the sides of the triangle. Each vertex of this polygon forms corners inside it.
Types of triangles
According to the size, angles of triangles, they are divided into such varieties as: Rectangular;
Acute-angled;
obtuse.
Right-angled triangles are triangles that have one right angle and the other two have acute angles.
Acute-angled triangles are those in which all of its angles are acute.
And if a triangle has one obtuse angle, and the other two angles are acute, then such a triangle belongs to obtuse angles.
Each of you is well aware that not all triangles have equal sides. And according to the length of its sides, triangles can be divided into:
Isosceles;
Equilateral;
Versatile.
Task: Draw different types of triangles. Give them a definition. What difference do you see between them?
Basic properties of triangles
Although these simple polygons may differ from each other in the size of the angles or sides, but in each triangle there are basic properties that are characteristic of this figure.
In any triangle:
The sum of all its angles is 180º.
If it belongs to equilateral, then each of its angles is equal to 60º.
An equilateral triangle has identical and equal angles to each other.
The smaller the side of the polygon, the smaller the angle opposite it, and vice versa, the larger angle is opposite the larger side.
If the sides are equal, then opposite them are equal angles, and vice versa.
If we take a triangle and extend its side, then in the end we will form an external angle. It is equal to the sum of the interior angles.
In any triangle, its side, no matter which one you choose, will still be less than the sum of the other 2 sides, but more than their difference:
1.a< b + c, a >b-c;
2.b< a + c, b >a-c;
3.c< a + b, c >a-b.
Exercise
The table shows the already known two angles of the triangle. Knowing the total sum of all the angles, find what the third angle of the triangle is equal to and enter in the table:
1. How many degrees does the third angle have?
2. What kind of triangles does it belong to?
Equivalence Triangles
I sign
II sign
III sign
Height, bisector and median of a triangle
The height of a triangle - the perpendicular drawn from the top of the figure to its opposite side, is called the height of the triangle. All heights of a triangle intersect at one point. The intersection point of all 3 altitudes of a triangle is its orthocenter.
A segment drawn from a given vertex and connecting it in the middle of the opposite side is the median. The medians, as well as the heights of a triangle, have one common point of intersection, the so-called center of gravity of the triangle or centroid.
The bisector of a triangle is a segment that connects the vertex of an angle and a point on the opposite side, and also divides this angle in half. All bisectors of a triangle intersect at one point, which is called the center of the circle inscribed in the triangle.
The segment that connects the midpoints of the 2 sides of the triangle is called the midline.
History reference
Such a figure as a triangle was known in ancient times. This figure and its properties were mentioned on Egyptian papyri four thousand years ago. A little later, thanks to the Pythagorean theorem and Heron's formula, the study of the property of a triangle moved to a higher level, but still, this happened more than two thousand years ago.
In the 15th-16th centuries, a lot of research began on the properties of a triangle, and as a result, such a science as planimetry arose, which was called the "New Triangle Geometry".
A scientist from Russia N. I. Lobachevsky made a huge contribution to the knowledge of the properties of triangles. His works later found application both in mathematics and in physics and cybernetics.
Thanks to the knowledge of the properties of triangles, such a science as trigonometry arose. It turned out to be necessary for a person in his practical needs, since its use is simply necessary when compiling maps, measuring areas, and even when designing various mechanisms.
What is the most famous triangle? This is, of course, the Bermuda Triangle! It got its name in the 50s because of the geographical location of the points (vertices of the triangle), within which, according to the existing theory, anomalies associated with it arose. The peaks of the Bermuda Triangle are Bermuda, Florida and Puerto Rico.
Assignment: What theories about the Bermuda Triangle have you heard?
Do you know that in Lobachevsky's theory, when adding the angles of a triangle, their sum always has a result less than 180º. In Riemannian geometry, the sum of all the angles of a triangle is greater than 180º, while in Euclid's writings it is equal to 180 degrees.
Homework
Solve a crossword puzzle on a given topic
Crossword questions:
1. What is the name of the perpendicular drawn from the vertex of the triangle to the straight line located on the opposite side?
2. How, in one word, can you call the sum of the lengths of the sides of a triangle?
3. Name a triangle whose two sides are equal?
4. Name a triangle that has an angle equal to 90°?
5. What is the name of the larger one of the sides of the triangle?
6. Name of the side of an isosceles triangle?
7. There are always three of them in any triangle.
8. What is the name of a triangle in which one of the angles exceeds 90 °?
9. The name of the segment connecting the top of our figure with the middle of the opposite side?
10. In a simple polygon ABC, the capital letter A is...?
11. What is the name of the segment that divides the angle of the triangle in half.
Questions about triangles:
1. Give a definition.
2. How many heights does it have?
3. How many bisectors does a triangle have?
4. What is its sum of angles?
5. What types of this simple polygon do you know?
6. Name the points of the triangles that are called wonderful.
7. What instrument can measure the angle?
8. If the hands of the clock show 21 hours. What angle do the hour hands form?
9. At what angle does a person turn if he is given the command "to the left", "around"?
10. What other definitions do you know that are associated with a figure that has three angles and three sides?
