Question

В ромбе одна диагональ длиннее другой на 8 см. сторона ромба 2 дм. определите площадь ромба

Answer 1

Диагонали пересекаются, образуется прямой угол. Ромб делится на 4 одинаковых прямоугольных треугольника.
2х - меньшая диагональ
2х+8 большая.
стороны треугольника равны:
х; х+4 и 20 см.
подставляем теорему пифагора, находим х.
$x^{2} + (x+4)^{2} = 20^{2} =400 \\ x^{2} + x^{2} +8x+16=400 \\ x^{2} +4x-192=0 \\ D = 4^2-4*1*(-192)=16+768=784 \\ x_{1}= \frac{-4-28}{2} =-16 \\ x_{2} = \frac{-4+28}{2} = 12$

длина не может быть отрицательной, поэтому меньшая половина диагонали равна 12
Меньшая диагональ равна 12*2=24
Большая диагональ равна 24+8=32
S=D*d/2=24*32/2=384
Ответ: 384 см кв.

Answer 2

Problem Analysis

We are given that one diagonal of a rhombus is 8 cm longer than the other diagonal. The length of one side of the rhombus is 2 dm. We need to determine the area of the rhombus.

Solution

To find the area of a rhombus, we can use the formula: Area = (diagonal1 * diagonal2) / 2.

Let's denote the length of the shorter diagonal as d1 and the length of the longer diagonal as d2.

From the given information, we know that d2 = d1 + 8 cm.

We also know that the length of one side of the rhombus is 2 dm, which is equal to 20 cm.

Since a rhombus has equal sides, we can use the Pythagorean theorem to find the length of the diagonals.

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

Using the Pythagorean theorem, we have:

s^2 = (d1/2)^2 + (d2/2)^2

Substituting the given values, we have:

20^2 = (d1/2)^2 + ((d1 + 8)/2)^2

Simplifying the equation, we get:

400 = (d1^2)/4 + (d1^2 + 16d1 + 64)/4

Multiplying both sides of the equation by 4, we get:

1600 = d1^2 + d1^2 + 16d1 + 64

Combining like terms, we have:

2d1^2 + 16d1 + 64 - 1600 = 0

Simplifying further, we get:

2d1^2 + 16d1 - 1536 = 0

Now we can solve this quadratic equation to find the value of d1.

Using the quadratic formula, we have:

d1 = (-b ± √(b^2 - 4ac)) / (2a)

Substituting the values, we get:

d1 = (-16 ± √(16^2 - 4 * 2 * -1536)) / (2 * 2)

Simplifying the equation, we get:

d1 = (-16 ± √(256 + 12288)) / 4

d1 = (-16 ± √12544) / 4

d1 = (-16 ± 112) / 4

Now we can calculate the two possible values of d1.

d1 = (-16 + 112) / 4 = 96 / 4 = 24 cm

d1 = (-16 - 112) / 4 = -128 / 4 = -32 cm

Since the length of a diagonal cannot be negative, we discard the negative value of d1.

Therefore, the length of the shorter diagonal (d1) is 24 cm and the length of the longer diagonal (d2) is 24 + 8 = 32 cm.

Now we can calculate the area of the rhombus using the formula:

Area = (d1 * d2) / 2 = (24 cm * 32 cm) / 2 = 384 cm^2

Answer

The area of the rhombus is 384 cm^2.