Question

Высота BM проведенная из вершины угла ромба АБСД образует со стороной АВ угол 30 грдусов, АМ = 4 см. Найдите длину диагонали АД Срочно плииииз

Answer 1

Высота ВМ  отсекает от ромба прямоугольный треугольник АВМ. Катет АМ противолежит углу 30º и равен половине гипотенузы АВ.

АВ=4*2=8 см

Сумма острых углов прямоугольного треугольника равна 90.

Т.к. угол АВМ=30º, угол ВАD=90-30=60

Все стороны ромба равны. ⇒АD=AB, треугольник ВАD равнобедренный с углом при А=60 ⇒угол АВD= углу ADB=(180-60):2=60

Треугольник АВD  равносторонний, BD=AB=AD=8 cм.

Диагональ ВD=8 см

Answer 2

Calculation of the Diagonal AD in a Rhombus

To find the length of the diagonal AD in a rhombus, we need to use the given information that the height BM, drawn from the vertex of angle ABS, forms a 30-degree angle with side AB and that AM is equal to 4 cm.

Let's proceed with the calculation:

1. We can start by drawing the rhombus and labeling the given information:


Here, AB represents the side of the rhombus, BM is the height drawn from the vertex of angle ABS, and AM is given as 4 cm.

2. Since angle AMB is 30 degrees, we can use trigonometric ratios to find the length of AB.

In a right triangle AMB, the side opposite the 30-degree angle is AB, and the side adjacent to the 30-degree angle is AM. We can use the tangent function to find AB:

tan(30 degrees) = AB / AM

Substituting the given values:

tan(30 degrees) = AB / 4 cm

Solving for AB:

AB = 4 cm * tan(30 degrees)

3. Now that we have the length of AB, we can find the length of the diagonal AD.

In a rhombus, the diagonals are perpendicular bisectors of each other. This means that the diagonal AD divides the rhombus into two congruent right triangles.

We can use the Pythagorean theorem to find the length of AD:

AD^2 = AB^2 + BD^2

Since the rhombus is symmetric, BD is equal to half the length of AB:

BD = AB / 2

Substituting the values:

AD^2 = AB^2 + (AB/2)^2

Simplifying:

AD^2 = AB^2 + AB^2/4

AD^2 = (5/4) * AB^2

Taking the square root of both sides:

AD = sqrt((5/4) * AB^2)

4. Now, we can substitute the value of AB that we found in step 2 into the equation for AD:

AD = sqrt((5/4) * (4 cm * tan(30 degrees))^2)

Simplifying:

AD = sqrt((5/4) * (4 cm)^2 * tan^2(30 degrees))

AD = sqrt(5) * 4 cm * tan(30 degrees)

Evaluating the trigonometric function:

AD = sqrt(5) * 4 cm * (sqrt(3)/3)

AD = 4 cm * sqrt(15/3)

AD = 4 cm * sqrt(5)

Therefore, the length of the diagonal AD in the rhombus is 4 cm * sqrt(5).

I hope this helps! Let me know if you have any further questions.