Solution of problem 1.1.6 from the collection of Kepe O.E.

Problem 1.1.6 from the collection of Kepe O.?. is as follows: “In triangle ABC, the altitudes AD and BE are drawn. Prove that the segment AB is equal to the sum of the segments AD and BE.”

This problem is a classic geometric problem that requires the application of knowledge about triangles and their properties. The solution to this problem is to use two properties of a triangle: the first property states that an altitude drawn to a side of the triangle is perpendicular to that side, and the second property states that an altitude drawn to a side of a triangle divides that side into two parts proportionally adjacent to her legs.

Using these properties, we can prove that the segment AB is indeed equal to the sum of the segments AD and BE. Solving this problem can be useful for students studying geometry, as well as for anyone interested in mathematics and its applications.

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Problem 1.1.6 from the collection of Kepe O.?. is as follows: “Prove that for any real numbers $a$ and $b$ the inequality $(a+b)^2\geqslant 4ab$ holds.”

This inequality is called the Cauchy-Bunyakovsky inequality for two numbers $a$ and $b$, and it is important in mathematics and its applications, especially in linear algebra and probability theory. The proof of this inequality is based on the properties of quadratic expressions and the properties of real numbers.

Solution to problem 1.1.6 from the collection of Kepe O.?. can be presented as a formal mathematical proof that uses various theorems and properties of real numbers and quadratic expressions, or as a brief text explanation that demonstrates the main ideas and steps needed to prove the Cauchy-Bunyakovsky inequality.

Solution to problem 1.1.6 from the collection of Kepe O.?. intended for students and schoolchildren studying physics and mechanics. In this problem, it is necessary to determine the modulus of force F2 if the resultant R = 10 H of two converging forces and the force F1 = 5 H, forming an angle with the Ox axis? = 60 о, as well as the angle between the resultant and the Ox axis, which is equal to? = 30 o.

To solve the problem, it is necessary to use the cosine theorem, which states that the square of the length of the resultant R is equal to the sum of the squares of the lengths of the forces F1 and F2, multiplied by the double product of these forces by the cosine of the angle between them:

R^2 = F1^2 + F2^2 + 2F1F2*cos(?)

Substituting the known values, we get:

10^2 = 5^2 + F2^2 + 25F2*cos(30)

Expressing F2, we get the answer:

F2 = (10^2 - 5^2)/(25cos(30)) = 6.64 H

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