Question

Сторона ромба равна 5 см, а длины диагоналей относятся как 4:3. Найти сумму диагоналей ромба

Answer 1

Если диагонали относятся 4:3, то и полудиагонали относятся 4:3.  Имеем прямоугольный треугольние с сторонами 3,4,5. Одни дигональ равна 3*2=6 Другая диагональ равн4*2=8 Сумма диагоналей 8+6=14 см

Answer 2

Finding the Sum of Diagonals of a Rhombus

To find the sum of the diagonals of a rhombus, we need to know the length of one side of the rhombus and the ratio between the lengths of the diagonals.

Given: - Length of one side of the rhombus = 5 cm - Ratio of the lengths of the diagonals = 4:3

Let's calculate the lengths of the diagonals and then find their sum.

Calculating the Lengths of the Diagonals

To calculate the lengths of the diagonals, we can use the fact that the diagonals of a rhombus bisect each other at right angles and divide the rhombus into four congruent right triangles.

Let's denote the lengths of the diagonals as D1 and D2.

Using the Pythagorean theorem, we can find the lengths of the diagonals in terms of the side length of the rhombus.

In each right triangle, the hypotenuse is one of the diagonals, and the legs are half the length of the side of the rhombus.

Let's denote half the length of the side of the rhombus as a/2.

Using the Pythagorean theorem, we have:

D1^2 = (a/2)^2 + (a/2)^2 D2^2 = (a/2)^2 + (a/2)^2

Simplifying the equations:

D1^2 = (a^2)/4 + (a^2)/4 = (2a^2)/4 = (a^2)/2 D2^2 = (a^2)/4 + (a^2)/4 = (2a^2)/4 = (a^2)/2

Taking the square root of both sides:

D1 = sqrt((a^2)/2) D2 = sqrt((a^2)/2)

Calculating the Sum of the Diagonals

Now that we have the lengths of the diagonals, we can calculate their sum.

Sum of the diagonals = D1 + D2

Substituting the values of D1 and D2:

Sum of the diagonals = sqrt((a^2)/2) + sqrt((a^2)/2)

Substituting the value of a = 5 cm:

Sum of the diagonals = sqrt((5^2)/2) + sqrt((5^2)/2)

Calculating the square roots:

Sum of the diagonals = sqrt(25/2) + sqrt(25/2)

Simplifying the square roots:

Sum of the diagonals = (5/√2) + (5/√2)

Adding the fractions:

Sum of the diagonals = (5 + 5)/√2

Simplifying the numerator:

Sum of the diagonals = 10/√2

To rationalize the denominator, we multiply the numerator and denominator by √2:

Sum of the diagonals = (10/√2) * (√2/√2)

Simplifying:

Sum of the diagonals = (10√2)/2

Final result:

Sum of the diagonals of the rhombus = 5√2 cm.

Please note that the exact numerical value of the sum of the diagonals depends on the value of √2, which is an irrational number.