Question

Периметр ромба равен 20 , а его высота равна 2,5. Найти диагональ ромба

Answer 1

Сторона ромба равна 20/4=5. 
Высота ромба BE равна 2.5.
Из треугольника ABE: sin∠A=2.5/5=1/2.
cos∠A=√(1-sin²∠A)=√3/2
Отсюда из треугольника ABD найдем первую диагональ BD по т. косинусов:
BD²=AB²+AD²-2*AB*AD*cos∠A=5²+5²-2*5*5*√3/2=50-25√3
BD=√(50-25√3)=5√(2-√3)=5(√6-√2)/2
Так как ∠A+∠B=180°, то cos∠B=-cos∠A=-√3/2.
Тогда из треугольника ABC по т. косинусов найдем вторую диагональ AC:
AC²=AB²+BC²-2AB*BC*cos∠B=5²+5²+2*5*5*√3/2=50+25√3
AC=√(50+25√3)=5√(2+√3)=5(√6+√2)/2

Answer 2

Perimeter of a Rhombus

To find the diagonal of a rhombus, we first need to calculate the perimeter of the rhombus. The perimeter of a rhombus is the sum of the lengths of all its sides.

Let's denote the perimeter of the rhombus as P and the length of each side as s.

The formula for the perimeter of a rhombus is:

P = 4s

In this case, we are given that the perimeter of the rhombus is 20. So we can set up the equation:

20 = 4s

Now, let's solve for s:

s = 20 / 4 = 5

Therefore, each side of the rhombus has a length of 5 units.

Height of the Rhombus

We are also given that the height of the rhombus is 2.5 units.

Finding the Diagonal of the Rhombus

To find the diagonal of the rhombus, we can use the formula:

Diagonal = 2 * (height / 2) * √(4 * (side length)^2 - (height)^2)

Substituting the given values:

Diagonal = 2 * (2.5 / 2) * √(4 * (5)^2 - (2.5)^2)

Simplifying the equation:

Diagonal = 2 * 1.25 * √(4 * 25 - 6.25)

Diagonal = 2.5 * √(100 - 6.25)

Diagonal = 2.5 * √(93.75)

Diagonal ≈ 2.5 * 9.686

Diagonal ≈ 24.215

Therefore, the diagonal of the rhombus is approximately 24.215 units.