The Pythagorean theorem states:
In a right triangle, the sum of the squares of the legs is equal to the square of the hypotenuse:
a 2 + b 2 = c 2,
- a And b– legs forming a right angle.
- With– hypotenuse of the triangle.
Formulas of the Pythagorean theorem
- a = \sqrt(c^(2) - b^(2))
- b = \sqrt (c^(2) - a^(2))
- c = \sqrt (a^(2) + b^(2))
Proof of the Pythagorean Theorem
Square right triangle calculated by the formula:
S = \frac(1)(2) ab
To calculate the area of an arbitrary triangle, the area formula is:
- p– semi-perimeter. p=\frac(1)(2)(a+b+c) ,
- r– radius of the inscribed circle. For a rectangle r=\frac(1)(2)(a+b-c).
Then we equate the right sides of both formulas for the area of the triangle:
\frac(1)(2) ab = \frac(1)(2)(a+b+c) \frac(1)(2)(a+b-c)
2 ab = (a+b+c) (a+b-c)
2 ab = \left((a+b)^(2) -c^(2) \right)
2 ab = a^(2)+2ab+b^(2)-c^(2)
0=a^(2)+b^(2)-c^(2)
c^(2) = a^(2)+b^(2)
Converse Pythagorean theorem:
If the square of one side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is right-angled. That is, for any triple of positive numbers a, b And c, such that
a 2 + b 2 = c 2,
there is a right triangle with legs a And b and hypotenuse c.
Pythagorean theorem- one of the fundamental theorems of Euclidean geometry, establishing the relationship between the sides of a right triangle. It was proven by the learned mathematician and philosopher Pythagoras.
The meaning of the theorem The point is that it can be used to prove other theorems and solve problems.
Additional material:
According to Van der Waerden, it is very likely that the ratio is general view was known in Babylon already around the 18th century BC. e.
Around 400 BC. BC, according to Proclus, Plato gave a method for finding Pythagorean triplets, combining algebra and geometry. Around 300 BC. e. The oldest axiomatic proof of the Pythagorean theorem appeared in Euclid's Elements.
Formulations
The basic formulation contains algebraic operations - in a right triangle, the lengths of which are equal a (\displaystyle a) And b (\displaystyle b), and the length of the hypotenuse is c (\displaystyle c), the following relation is satisfied:
.An equivalent geometric formulation is also possible, resorting to the concept of area of a figure: 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. The theorem is formulated in this form in Euclid’s Elements.
Converse Pythagorean theorem- a statement about the rectangularity of any triangle, the lengths of the sides of which are related by the relation a 2 + b 2 = c 2 (\displaystyle a^(2)+b^(2)=c^(2)). As a consequence, for every triple of positive numbers a (\displaystyle a), b (\displaystyle b) And c (\displaystyle c), such that a 2 + b 2 = c 2 (\displaystyle a^(2)+b^(2)=c^(2)), there is a right triangle with legs a (\displaystyle a) And b (\displaystyle b) and hypotenuse c (\displaystyle c).
Proof
IN scientific literature At least 400 proofs of the Pythagorean theorem have been recorded, which is explained both by its fundamental significance for geometry and by the elementary nature of the result. The main directions of proofs are: algebraic use of relations between the elements of a triangle (for example, the popular method of similarity), the method of areas, there are also various exotic proofs (for example, using differential equations).
Through similar triangles
Euclid's classical proof aims to establish the equality of areas between rectangles formed by dissecting a square above the hypotenuse with a height of right angle with squares over the legs.
The construction used for the proof is as follows: for a right triangle with a right angle C (\displaystyle C), squares over the legs and and squares over the hypotenuse A B I K (\displaystyle ABIK) height is being built CH and the ray that continues it s (\displaystyle s), dividing the square above the hypotenuse into two rectangles and . The proof aims to establish the equality of the areas of the rectangle A H J K (\displaystyle AHJK) with a square over the leg A C (\displaystyle AC); the equality of the areas of the second rectangle, constituting the square above the hypotenuse, and the rectangle above the other leg is established in a similar way.
Equality of areas of a rectangle A H J K (\displaystyle AHJK) And A C E D (\displaystyle ACED) is established through the congruence of triangles △ A C K (\displaystyle \triangle ACK) And △ A B D (\displaystyle \triangle ABD), the area of each of which is equal to half the area of the squares A H J K (\displaystyle AHJK) And A C E D (\displaystyle ACED) accordingly, in connection with the following property: the area of a triangle is equal to half the area of a rectangle if the figures have a common side, and the height of the triangle to the common side is the other side of the rectangle. The congruence of triangles follows from the equality of two sides (sides of squares) and the angle between them (composed of a right angle and an angle at A (\displaystyle A).
Thus, the proof establishes that the area of a square above the hypotenuse, composed of rectangles A H J K (\displaystyle AHJK) And B H J I (\displaystyle BHJI), is equal to the sum of the areas of the squares over the legs.
Proof of Leonardo da Vinci
The area method also includes a proof found by Leonardo da Vinci. Let a right triangle be given △ A B C (\displaystyle \triangle ABC) with right angle C (\displaystyle C) and squares A C E D (\displaystyle ACED), B C F G (\displaystyle BCFG) And A B H J (\displaystyle ABHJ)(see picture). In this proof on the side HJ (\displaystyle HJ) the latter, a triangle is constructed on the outer side, congruent △ A B C (\displaystyle \triangle ABC), moreover, reflected both relative to the hypotenuse and relative to the height to it (that is, J I = B C (\displaystyle JI=BC) And H I = A C (\displaystyle HI=AC)). Straight C I (\displaystyle CI) splits the square built on the hypotenuse into two equal parts, since triangles △ A B C (\displaystyle \triangle ABC) And △ J H I (\displaystyle \triangle JHI) equal in construction. The proof establishes the congruence of quadrilaterals C A J I (\displaystyle CAJI) And D A B G (\displaystyle DABG), the area of each of which turns out to be, on the one hand, equal to the sum of half the areas of the squares on the legs and the area of the original triangle, on the other hand, half the area of the square on the hypotenuse plus the area of the original triangle. In total, half the sum of the areas of the squares over the legs is equal to half the area of the square over the hypotenuse, which is equivalent to the geometric formulation of the Pythagorean theorem.
Proof by the infinitesimal method
There are several proofs using the technique of differential equations. In particular, Hardy is credited with a proof using infinitesimal increments of legs a (\displaystyle a) And b (\displaystyle b) and hypotenuse c (\displaystyle c), and preserving similarity with the original rectangle, that is, ensuring the fulfillment of the following differential relations:
d a d c = c a (\displaystyle (\frac (da)(dc))=(\frac (c)(a))), d b d c = c b (\displaystyle (\frac (db)(dc))=(\frac (c)(b))).Using the method of separating variables, one can derive from them differential equation c d c = a d a + b d b (\displaystyle c\ dc=a\,da+b\,db), whose integration gives the relation c 2 = a 2 + b 2 + C o n s t (\displaystyle c^(2)=a^(2)+b^(2)+\mathrm (Const) ). Application of initial conditions a = b = c = 0 (\displaystyle a=b=c=0) defines the constant as 0, which results in the statement of the theorem.
The quadratic dependence in the final formula appears due to the linear proportionality between the sides of the triangle and the increments, while the sum is associated with independent contributions from the increment of different legs.
Variations and generalizations
Similar geometric shapes on three sides
An important geometric generalization of the Pythagorean theorem was given by Euclid in the Elements, moving from the areas of squares on the sides to the areas of arbitrary similar geometric shapes: the sum of the areas of such figures built on the legs will be equal to the area of a similar figure built on the hypotenuse.
The main idea of this generalization is that the area of such a geometric figure is proportional to the square of any of its linear dimensions and, in particular, to the square of the length of any side. Therefore, for similar figures with areas A (\displaystyle A), B (\displaystyle B) And C (\displaystyle C), built on legs with lengths a (\displaystyle a) And b (\displaystyle b) and hypotenuse c (\displaystyle c) Accordingly, the following relation holds:
A a 2 = B b 2 = C c 2 ⇒ A + B = a 2 c 2 C + b 2 c 2 C (\displaystyle (\frac (A)(a^(2)))=(\frac (B )(b^(2)))=(\frac (C)(c^(2)))\,\Rightarrow \,A+B=(\frac (a^(2))(c^(2) ))C+(\frac (b^(2))(c^(2)))C).Since according to the Pythagorean theorem a 2 + b 2 = c 2 (\displaystyle a^(2)+b^(2)=c^(2)), then done.
In addition, if it is possible to prove without invoking the Pythagorean theorem that the areas of three similar geometric figures on the sides of a right triangle satisfy the relation A + B = C (\displaystyle A+B=C), then using the reverse of the proof of Euclid's generalization, one can derive a proof of the Pythagorean theorem. For example, if on the hypotenuse we construct a right triangle congruent with the initial one with an area C (\displaystyle C), and on the sides - two similar right-angled triangles with areas A (\displaystyle A) And B (\displaystyle B), then it turns out that triangles on the sides are formed as a result of dividing the initial triangle by its height, that is, the sum of the two smaller areas of the triangles is equal to the area of the third, thus A + B = C (\displaystyle A+B=C) and by applying the relation to similar figures, the Pythagorean theorem is derived.
Cosine theorem
The Pythagorean theorem is special case the more general cosine theorem, which relates the lengths of the sides in an arbitrary triangle:
a 2 + b 2 − 2 a b cos θ = c 2 (\displaystyle a^(2)+b^(2)-2ab\cos (\theta )=c^(2)),where is the angle between the sides a (\displaystyle a) And b (\displaystyle b). If the angle is 90°, then cos θ = 0 (\displaystyle \cos \theta =0), and the formula simplifies to the usual Pythagorean theorem.
Free Triangle
There is a generalization of the Pythagorean theorem to an arbitrary triangle, operating solely on the ratio of the lengths of the sides, it is believed that it was first established by the Sabian astronomer Thabit ibn Qurra. In it, for an arbitrary triangle with sides, an isosceles triangle with a base on the side fits into it c (\displaystyle c), the vertex coinciding with the vertex of the original triangle, opposite the side c (\displaystyle c) and corners at the base, equal to the angle θ (\displaystyle \theta ), opposite side c (\displaystyle c). As a result, two triangles are formed, similar to the original one: the first - with sides a (\displaystyle a), the side farthest from it of the inscribed isosceles triangle, And r (\displaystyle r)- side parts c (\displaystyle c); the second - symmetrically to it from the side b (\displaystyle b) with the side s (\displaystyle s)- the corresponding part of the side c (\displaystyle c). As a result, the following relation is satisfied:
a 2 + b 2 = c (r + s) (\displaystyle a^(2)+b^(2)=c(r+s)),degenerating into the Pythagorean theorem at θ = π / 2 (\displaystyle \theta =\pi /2). The relationship is a consequence of the similarity of the formed triangles:
c a = a r , c b = b s ⇒ c r + c s = a 2 + b 2 (\displaystyle (\frac (c)(a))=(\frac (a)(r)),\,(\frac (c) (b))=(\frac (b)(s))\,\Rightarrow \,cr+cs=a^(2)+b^(2)).Pappus's theorem on areas
Non-Euclidean geometry
The Pythagorean theorem is derived from the axioms of Euclidean geometry and is not valid for non-Euclidean geometry - the fulfillment of the Pythagorean theorem is equivalent to the Euclidian parallelism postulate.
In non-Euclidean geometry, the relationship between the sides of a right triangle will necessarily be in a form different from the Pythagorean theorem. For example, in spherical geometry, all three sides of a right triangle, which bound the octant of the unit sphere, have a length π / 2 (\displaystyle \pi /2), which contradicts the Pythagorean theorem.
Moreover, the Pythagorean theorem is valid in hyperbolic and elliptic geometry if the requirement that the triangle is rectangular is replaced by the condition that the sum of two angles of the triangle must be equal to the third.
Spherical geometry
For any right triangle on a sphere with radius R (\displaystyle R)(for example, if the angle in a triangle is right) with sides a , b , c (\displaystyle a,b,c) the relationship between the sides is:
cos (c R) = cos (a R) ⋅ cos (b R) (\displaystyle \cos \left((\frac (c)(R))\right)=\cos \left((\frac (a)(R))\right)\cdot \cos \left((\frac (b)(R))\right)).This equality can be derived as a special case of the spherical cosine theorem, which is valid for all spherical triangles:
cos (c R) = cos (a R) ⋅ cos (b R) + sin (a R) ⋅ sin (b R) ⋅ cos γ (\displaystyle \cos \left((\frac ( c)(R))\right)=\cos \left((\frac (a)(R))\right)\cdot \cos \left((\frac (b)(R))\right)+\ sin \left((\frac (a)(R))\right)\cdot \sin \left((\frac (b)(R))\right)\cdot \cos \gamma ). ch c = ch a ⋅ ch b (\displaystyle \operatorname (ch) c=\operatorname (ch) a\cdot \operatorname (ch) b),Where ch (\displaystyle \operatorname (ch) )- hyperbolic cosine. This formula is a special case of the hyperbolic cosine theorem, which is valid for all triangles:
ch c = ch a ⋅ ch b − sh a ⋅ sh b ⋅ cos γ (\displaystyle \operatorname (ch) c=\operatorname (ch) a\cdot \operatorname (ch) b-\operatorname (sh) a\cdot \operatorname (sh) b\cdot \cos \gamma ),Where γ (\displaystyle \gamma )- an angle whose vertex is opposite to the side c (\displaystyle c).
Using the Taylor series for the hyperbolic cosine ( ch x ≈ 1 + x 2 / 2 (\displaystyle \operatorname (ch) x\approx 1+x^(2)/2)) it can be shown that if a hyperbolic triangle decreases (that is, when a (\displaystyle a), b (\displaystyle b) And c (\displaystyle c) tend to zero), then the hyperbolic relations in a right triangle approach the relation of the classical Pythagorean theorem.
Application
Distance in two-dimensional rectangular systems
The most important application of the Pythagorean theorem is determining the distance between two points in a rectangular coordinate system: distance s (\displaystyle s) between points with coordinates (a , b) (\displaystyle (a,b)) And (c , d) (\displaystyle (c,d)) equals:
s = (a − c) 2 + (b − d) 2 (\displaystyle s=(\sqrt ((a-c)^(2)+(b-d)^(2)))).For complex numbers The Pythagorean theorem gives a natural formula for finding the modulus of a complex number - for z = x + y i (\displaystyle z=x+yi) it is equal to the length
It is remarkable that the property specified in the Pythagorean theorem is a characteristic property of a right triangle. This follows from the theorem converse to the Pythagorean theorem.
Theorem: If the square of one side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is right-angled.
Heron's formula
Let us derive a formula expressing the plane of a triangle in terms of the lengths of its sides. This formula is associated with the name of Heron of Alexandria - an ancient Greek mathematician and mechanic who probably lived in the 1st century AD. Heron paid much attention to the practical applications of geometry.
Theorem. The area S of a triangle whose sides are equal to a, b, c is calculated by the formula S=, where p is the semi-perimeter of the triangle.
Proof.
Given: ?ABC, AB= c, BC= a, AC= b. Angles A and B are acute. CH - height.
Prove:
Proof:
Let's consider triangle ABC, in which AB=c, BC=a, AC=b. Every triangle has at least two acute angles. Let A and B be sharp corners triangle ABC. Then the base H of altitude CH of the triangle lies on side AB. Let us introduce the following notation: CH = h, AH=y, HB=x. by the Pythagorean theorem a 2 - x 2 = h 2 =b 2 -y 2, whence
Y 2 - x 2 = b 2 - a 2, or (y - x) (y + x) = b 2 - a 2, and since y + x = c, then y- x = (b2 - a2).
Adding the last two equalities, we get:
2y = +c, whence
y=, and, therefore, h 2 = b 2 -y 2 =(b - y)(b+y)=
Therefore, h = .
Subject: Theorem, converse of the theorem Pythagoras.
Lesson objectives: 1) consider the theorem converse to the Pythagorean theorem; its application in the process of problem solving; consolidate the Pythagorean theorem and improve problem solving skills for its application;
2) develop logical thinking, creative search, cognitive interest;
3) to cultivate in students a responsible attitude to learning and a culture of mathematical speech.
Lesson type. A lesson in learning new knowledge.
Lesson progress
ІІ. Update knowledge
Lesson for mewouldI wantedstart with a quatrain.
Yes, the path of knowledge is not smooth
But we know school years,
There are more mysteries than answers,
And there is no limit to the search!
So, in the last lesson you learned the Pythagorean theorem. Questions:
The Pythagorean theorem is true for which figure?
Which triangle is called a right triangle?
State the Pythagorean theorem.
How can the Pythagorean theorem be written for each triangle?
Which triangles are called equal?
Formulate the criteria for the equality of triangles?
Now let's do a little independent work:
Solving problems using drawings.
№1
(1 b.) Find: AB.
№2
(1 b.) Find: VS.
№3
( 2 b.)Find: AC
№4
(1 point)Find: AC
№5 Given by: ABCDrhombus
(2 b.) AB = 13 cm
AC = 10 cm
Find: BD
Self-test No. 1. 5
№2. 5
№3. 16
№4. 13
№5. 24
ІІІ. Studying new material.
The ancient Egyptians built right angles on the ground in this way: they divided the rope into 12 knots equal parts, tied its ends, after which the rope was stretched on the ground so that a triangle was formed with sides of 3, 4 and 5 divisions. The angle of the triangle that lay opposite the side with 5 divisions was right.
Can you explain the correctness of this judgment?
As a result of searching for an answer to the question, students should understand that from a mathematical point of view the question is posed: will the triangle be right-angled?
We pose a problem: how to determine, without making measurements, whether a triangle with given sides will be rectangular. Solving this problem is the goal of the lesson.
Write down the topic of the lesson.
Theorem. If the sum of the squares of two sides of a triangle is equal to the square of the third side, then the triangle is right-angled.
Prove the theorem independently (make a proof plan using the textbook).
From this theorem it follows that a triangle with sides 3, 4, 5 is right-angled (Egyptian).
In general, numbers for which equality holds , are called Pythagorean triplets. And triangles whose side lengths are expressed by Pythagorean triplets (6, 8, 10) are Pythagorean triangles.
Consolidation.
Because , then a triangle with sides 12, 13, 5 is not right-angled.
Because , then a triangle with sides 1, 5, 6 is right-angled.
№ 430 (a, b, c)
( - is not)
