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 equal to the sum 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 - ancient Greek mathematician and a 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 acute angles of 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 = .
Pythagorean theorem- one of the fundamental theorems of Euclidean geometry, establishing the relation
between the sides of a right triangle.
It is believed that it was proven by the Greek mathematician Pythagoras, after whom it was named.
Geometric formulation of the Pythagorean theorem.
The theorem was originally formulated as follows:
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 legs.
Algebraic formulation of the Pythagorean theorem.
In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.
That is, denoting the length of the hypotenuse of the triangle by c, and the lengths of the legs through a And b:
Both formulations Pythagorean theorem are equivalent, but the second formulation is more elementary, it does not
requires the concept of area. That is, the second statement can be verified without knowing anything about the area and
by measuring only the lengths of the sides of a right triangle.
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
right triangle.
Or, in other words:
For every triple of positive numbers a, b And c, such that
there is a right triangle with legs a And b and hypotenuse c.
Pythagorean theorem for an isosceles triangle.
Pythagorean theorem for an equilateral triangle.
Proofs of the Pythagorean theorem.
Currently, 367 proofs of this theorem have been recorded in the scientific literature. Probably the theorem
Pythagoras is the only theorem with such an impressive number of proofs. Such diversity
can only be explained by the fundamental significance of the theorem for geometry.
Of course, conceptually all of them can be divided into a small number of classes. The most famous of them:
proof area method, axiomatic And exotic evidence(For example,
by using differential equations).
1. Proof of the Pythagorean theorem using similar triangles.
The following proof of the algebraic formulation is the simplest of the proofs constructed
directly from the axioms. In particular, it does not use the concept of area of a figure.
Let ABC there is a right triangle with a right angle C. Let's draw the height from C and denote
its foundation through H.
Triangle ACH similar to a triangle AB C at two corners. Likewise, triangle CBH similar ABC.
By introducing the notation:
we get:
,
which corresponds to -
Folded a 2 and b 2, we get:
or , which is what needed to be proven.
2. Proof of the Pythagorean theorem using the area method.
The proofs below, despite their apparent simplicity, are not so simple at all. All of them
use properties of area, the proofs of which are more complex than the proof of the Pythagorean theorem itself.
- Proof through equicomplementarity.
Let's arrange four equal rectangular
triangle as shown in the figure
on right.
Quadrangle with sides c- square,
since the sum of two acute angles is 90°, and
unfolded angle - 180°.
The area of the entire figure is equal, on the one hand,
area of a square with side ( a+b), and on the other hand, the sum of areas four triangles And
Q.E.D.
3. Proof of the Pythagorean theorem by the infinitesimal method.
Looking at the drawing shown in the figure and
watching the side changea, we can
write the following relation for infinitely
small side incrementsWith And a(using similarity
triangles):
Using the variable separation method, we find:
A more general expression for the change in the hypotenuse in the case of increments on both sides:
Integrating this equation and using the initial conditions, we obtain:
Thus we arrive at the desired answer:
As is easy to see, 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 related to the independent
contributions from the increment of different legs.
A simpler proof can be obtained if we assume that one of the legs does not experience an increase
(V in this case leg b). Then for the integration constant we obtain:
Consideration of topics school curriculum with the help of video tutorials is in a convenient way studying and mastering the material. The video helps to concentrate students’ attention on the main theoretical principles and not to miss important details. If necessary, students can always listen to the video lesson again or go back several topics.
This video lesson for 8th grade will help students learn new topic in geometry.
In the previous topic, we studied the Pythagorean theorem and analyzed its proof.
There is also a theorem that is known as the inverse Pythagorean theorem. Let's take a closer look at it.
Theorem. A triangle is right-angled if it holds the following equality: the value of one side of the triangle squared is the same as the sum of the other two sides squared.
Proof. Suppose we are given a triangle ABC, in which the equality AB 2 = CA 2 + CB 2 holds. It is necessary to prove that angle C is 90 degrees. Consider a triangle A 1 B 1 C 1 in which angle C 1 is equal to 90 degrees, side C 1 A 1 is equal to CA and side B 1 C 1 is equal to BC.
Applying the Pythagorean theorem, we write the ratio of the sides in the triangle A 1 C 1 B 1: A 1 B 1 2 = C 1 A 1 2 + C 1 B 1 2. By replacing the expression with equal sides, we get A 1 B 1 2 = CA 2 + CB 2.
From the conditions of the theorem we know that AB 2 = CA 2 + CB 2. Then we can write A 1 B 1 2 = AB 2, from which it follows that A 1 B 1 = AB.
We found that in triangles ABC and A 1 B 1 C 1 three sides are equal: A 1 C 1 = AC, B 1 C 1 = BC, A 1 B 1 = AB. So these triangles are equal. From the equality of triangles it follows that angle C equal to angle From 1 and accordingly equals 90 degrees. We have determined that triangle ABC is right-angled and its angle C is 90 degrees. We have proven this theorem.
Next, the author gives an example. Suppose we are given an arbitrary triangle. The sizes of its sides are known: 5, 4 and 3 units. Let's check the statement from the theorem inverse to the Pythagorean theorem: 5 2 = 3 2 + 4 2. The statement is true, which means this triangle is right-angled.
In the following examples, triangles will also be right triangles if their sides are equal:
5, 12, 13 units; the equality 13 2 = 5 2 + 12 2 is true;
8, 15, 17 units; the equality 17 2 = 8 2 + 15 2 is true;
7, 24, 25 units; the equality 25 2 = 7 2 + 24 2 is true.
The concept of a Pythagorean triangle is known. This is a right triangle whose sides are equal to whole numbers. If the legs of the Pythagorean triangle are denoted by a and c, and the hypotenuse by b, then the values of the sides of this triangle can be written using the following formulas:
b = k x (m 2 - n 2)
c = k x (m 2 + n 2)
where m, n, k are any integers, and the value of m is greater than the value of n.
Interesting fact: a triangle with sides 5, 4 and 3 is also called Egyptian triangle, such a triangle was known back in Ancient Egypt.
In this video lesson we learned the theorem converse to the Pythagorean theorem. We examined the evidence in detail. Students also learned which triangles are called Pythagorean triangles.
Students can easily familiarize themselves with the topic “The Inverse Theorem of Pythagoras” on their own with the help of this video lesson.
Subject: The theorem inverse to the Pythagorean theorem.
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.
During the classes
ІІ. Update knowledge
Lesson for mewouldI wantedstart with a quatrain.
Yes, the path of knowledge is not smooth
But we know from our 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 inD
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 equal parts with knots, 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 the 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)
