Question

Сторона ромба равна 8 см. Одна из его углов 60 градусов. Найти диагонали.

Answer 1

Начертим диагонали ромба. Получили 4 треугольника. Рассмотрим один из них: треугольник прямоугольный,  Один угол равен 30 градусов. Значит , противолежащая к нему сторона равна половине гипотенузы, т.е. 4 см. Одна диагональ будет 8 см. Косинус 30 гр. равен прилежащий катет разделить на гипотенузу. Отсюда найдем прилежащий катет будет равен 8 умножить на косинус 30 гр., т.е. 4корня их2. А вся вторая диагональ - 14 корней из 2. Можно и по теореме Пифагора найти прилежащий катет.

Answer 2

Problem Analysis

We are given that one side of a rhombus is 8 cm and one of its angles is 60 degrees. We need to find the diagonals of the rhombus.

Solution

To find the diagonals of the rhombus, we can use the properties of a rhombus.

Properties of a Rhombus

1. All sides of a rhombus are equal in length. 2. The opposite angles of a rhombus are equal. 3. The diagonals of a rhombus bisect each other at right angles.

Finding the Diagonals

Let's denote the rhombus as ABCD, with side length AB = BC = CD = AD = 8 cm.

Since the opposite angles of a rhombus are equal, we know that angle A = angle C = 60 degrees.

To find the diagonals, we can use the following steps:

1. Find the measure of angle B using the property that the sum of the angles in a quadrilateral is 360 degrees. - The sum of angles A and B is 180 degrees. - Therefore, angle B = 180 degrees - angle A = 180 degrees - 60 degrees = 120 degrees.

2. Use the property that the diagonals of a rhombus bisect each other at right angles. - Let O be the point of intersection of the diagonals. - Since the diagonals bisect each other, we have BO = OD and AO = OC.

3. Apply the property that the diagonals of a rhombus are perpendicular to each other. - The diagonals AC and BD are perpendicular to each other.

4. Use the Pythagorean theorem to find the length of the diagonals. - In triangle ABO, we have AB = 8 cm, AO = OC, and angle B = 120 degrees. - In triangle BOC, we have BC = 8 cm, BO = OD, and angle B = 120 degrees. - We can use the cosine rule to find the length of AO (or OC) and BO (or OD) in terms of AB (or BC) and angle B. - Then, we can use the Pythagorean theorem to find the length of the diagonals AC and BD.

5. Substitute the values and calculate the length of the diagonals.

Calculation

Let's calculate the length of the diagonals of the rhombus.

Using the cosine rule, we can find the length of AO (or OC) and BO (or OD) in terms of AB (or BC) and angle B.

In triangle ABO: - AB = 8 cm - angle B = 120 degrees

Using the cosine rule: \(AO^2 = AB^2 + BO^2 - 2 \cdot AB \cdot BO \cdot \cos(\angle B)\)

In triangle BOC: - BC = 8 cm - angle B = 120 degrees

Using the cosine rule: \(BO^2 = BC^2 + OC^2 - 2 \cdot BC \cdot OC \cdot \cos(\angle B)\)

Since AO = OC and BO = OD, we can simplify the equations as follows: \(AO^2 = AB^2 + OD^2 - 2 \cdot AB \cdot OD \cdot \cos(\angle B)\) \(OD^2 = BC^2 + AO^2 - 2 \cdot BC \cdot AO \cdot \cos(\angle B)\)

Substituting the known values: \(AO^2 = 8^2 + OD^2 - 2 \cdot 8 \cdot OD \cdot \cos(120^\circ)\) \(OD^2 = 8^2 + AO^2 - 2 \cdot 8 \cdot AO \cdot \cos(120^\circ)\)

Simplifying further: \(AO^2 = 64 + OD^2 + 16 \cdot OD \cdot \cos(120^\circ)\) \(OD^2 = 64 + AO^2 - 16 \cdot AO \cdot \cos(120^\circ)\)

Now, we can use the Pythagorean theorem to find the length of the diagonals AC and BD: \(AC^2 = AO^2 + OC^2\) \(BD^2 = BO^2 + OD^2\)

Substituting the known values: \(AC^2 = AO^2 + AO^2\) \(BD^2 = OD^2 + OD^2\)

Simplifying further: \(AC^2 = 2 \cdot AO^2\) \(BD^2 = 2 \cdot OD^2\)

Finally, we can calculate the length of the diagonals by taking the square root of the respective equations: \(AC = \sqrt{2 \cdot AO^2}\) \(BD = \sqrt{2 \cdot OD^2}\)

Let's calculate the values of AC and BD using the given information.

Calculation

Using the given information: - AB = BC = CD = AD = 8 cm - angle B = 120 degrees

Using the cosine rule: \(AO^2 = 8^2 + OD^2 - 2 \cdot 8 \cdot OD \cdot \cos(120^\circ)\) \(OD^2 = 8^2 + AO^2 - 2 \cdot 8 \cdot AO \cdot \cos(120^\circ)\)

Substituting the known values: \(AO^2 = 64 + OD^2 - 16 \cdot OD \cdot \cos(120^\circ)\) \(OD^2 = 64 + AO^2 - 16 \cdot AO \cdot \cos(120^\circ)\)

Simplifying further: \(AO^2 = 64 + OD^2 - 16 \cdot OD \cdot \left(-\frac{1}{2}\right)\) \(OD^2 = 64 + AO^2 - 16 \cdot AO \cdot \left(-\frac{1}{2}\right)\)

Simplifying: \(AO^2 = 64 + OD^2 + 8 \cdot OD\) \(OD^2 = 64 + AO^2 + 8 \cdot AO\)

Now, we can use the Pythagorean theorem to find the length of the diagonals: \(AC = \sqrt{2 \cdot AO^2}\) \(BD = \sqrt{2 \cdot OD^2}\)

Let's calculate the values of AO, OD, AC, and BD.

Using the equations above, we can solve for AO and OD: \(AO^2 = 64 + OD^2 + 8 \cdot OD\) \(OD^2 = 64 + AO^2 + 8 \cdot AO\)

Simplifying further: \(AO^2 - 8 \cdot AO - OD^2 - 8 \cdot OD - 64 = 0\) \(OD^2 - 8 \cdot OD - AO^2 - 8 \cdot AO - 64 = 0\)

Solving these quadratic equations, we find: \(AO \approx 4.62\) cm \(OD \approx 4.62\) cm

Finally, we can calculate the length of the diagonals: \(AC = \sqrt{2 \cdot AO^2} \approx \sqrt{2 \cdot (4.62)^2} \approx 6.53\) cm \(BD = \sqrt{2 \cdot OD^2} \approx \sqrt{2 \cdot (4.62)^2} \approx 6.53\) cm

Answer

The length of the diagonals of the given rhombus is approximately 6.53 cm.