Question

3. В параллелограмме ABCD угол А равен 60". Высота BE делит сторону AD на две равные части. Найдите длину диагонали BD, если периметр параллелограмма равен 48 см. Помогите пожалуйста СОЧ срочнооооо!!!!​

Answer 1

Ответ:

8 см

Объяснение:

Диагональ делит параллелограмм на 2 равных треугольника.

Р(АВD)=P(BCD)=48:2=24 cм

ΔАВD - равнобедренный, т.к. АЕ=ЕD, а ВЕ - высота и медиана

∠А=∠D=60°, ∠ABD=180-60-60=60°, значит, ΔАВD - равносторонний.

BD=24:3=6 см

Answer 2

Problem Analysis

We are given a parallelogram ABCD, where angle A is 60 degrees. The height BE divides the side AD into two equal parts. We need to find the length of diagonal BD, given that the perimeter of the parallelogram is 48 cm.

Solution

To solve this problem, we can use the properties of a parallelogram and the given information.

Let's denote the length of side AD as x. Since the height BE divides AD into two equal parts, each part will have a length of x/2.

The perimeter of a parallelogram is given by the sum of the lengths of all four sides. In this case, the perimeter is 48 cm, so we can write:

2 * (AB + BC) = 48

Since opposite sides of a parallelogram are equal in length, we can simplify the equation to:

2 * (AB + AD) = 48

Since AB and AD are opposite sides, they are equal in length. Let's denote their length as y. Now we have:

2 * (y + y) = 48

Simplifying further:

4y = 48

Dividing both sides by 4:

y = 12

So, the length of sides AB and AD is 12 cm.

Now, let's consider triangle ABE. We know that angle A is 60 degrees, and BE is the height of the parallelogram. Since AB and AD are equal in length, triangle ABE is an equilateral triangle. In an equilateral triangle, all angles are 60 degrees.

Therefore, angle BAE is also 60 degrees. Since the sum of angles in a triangle is 180 degrees, angle ABE is 180 - 60 - 60 = 60 degrees.

Now, we can use the sine rule to find the length of BE. In triangle ABE, we have:

sin(A) / AB = sin(B) / BE

Substituting the values:

sin(60) / 12 = sin(60) / BE

Simplifying:

BE = 12 / sin(60)

Using the value of sin(60) = √3 / 2:

BE = 12 / (√3 / 2) = 24 / √3 = (24√3) / 3 = 8√3

So, the length of BE is 8√3 cm.

Now, let's consider triangle BDE. We know that angle BDE is 90 degrees, and we have the lengths of BE and BD. We can use the Pythagorean theorem to find the length of BD.

According to the Pythagorean theorem:

BD^2 = BE^2 + DE^2

Since BE is the height of the parallelogram, it is equal to the length of AD, which is 12 cm. Let's denote the length of DE as d.

Substituting the values:

BD^2 = (8√3)^2 + d^2

Simplifying:

BD^2 = 192 + d^2

Since BD is the diagonal of the parallelogram, it is equal to the length of AC. We know that AC is equal to the perimeter of the parallelogram minus the sum of the lengths of AB and BC:

AC = 48 - (12 + 12) = 48 - 24 = 24 cm

Therefore, BD is also 24 cm.

Substituting the values:

24^2 = 192 + d^2

576 = 192 + d^2

Subtracting 192 from both sides:

384 = d^2

Taking the square root of both sides:

d = √384 = √(16 * 24) = 4√24 = 4√(4 * 6) = 8√6

So, the length of DE is 8√6 cm.

Now, we can find the length of BD by substituting the values into the Pythagorean theorem:

BD^2 = (8√3)^2 + (8√6)^2

Simplifying:

BD^2 = 192 + 384 = 576

Taking the square root of both sides:

BD = √576 = 24

Therefore, the length of diagonal BD is 24 cm.

Answer

The length of diagonal BD in the parallelogram ABCD is 24 cm.