Question

Пожалуйста, решите! отдаю последние баллы, всё потратила, никто не решает! Радиус вписанной в ромб окружности равен 5, а один из углов ромба равен 60°. Найти длину большей диагонали ромба.

Answer 1

Ромб ABCD; ∠A=60°; O - точка пересечения диагоналей, она же  -центр вписанной окружности. OE - высота, опущенная на сторону AD; равна радиусу, то есть 5. AO делит угол BAD пополам⇒∠OAD=30°⇒AD - катет, лежащий против угла в 30°⇒он равен половине гипотенузы AO, то есть AO=10⇒AC=2·AO=20. Поскольку диагональ AC выходит из меньшего угла ромба⇒она является большей диагональю.

Ответ: 20

Answer 2

Given Information:

- The radius of the circle inscribed in the rhombus is 5. - One of the angles of the rhombus is 60°.

To Find:

The length of the longer diagonal of the rhombus.

Approach:

To find the length of the longer diagonal of the rhombus, we can use the properties of the rhombus and the circle inscribed in it.

Solution:

Let's consider the rhombus and the inscribed circle as shown below:

      A
     / \
    /   \
   /     \
  /       \
D --------- B
  \       /
   \     /
    \   /
     \ /
      C

- Let AB be the longer diagonal of the rhombus, and let AC be the shorter diagonal. - Let O be the center of the circle inscribed in the rhombus. - Let M be the midpoint of AC.

Since the circle is inscribed in the rhombus, the center of the circle (O) coincides with the intersection point of the diagonals (AC and BD).

In a rhombus, the diagonals are perpendicular bisectors of each other. Therefore, OM is perpendicular to AC and BM is perpendicular to AC.

Let's consider triangle OMB. In triangle OMB, OM is the height, and BM is the base. The angle between OM and BM is 60°, as given.

Since OM is perpendicular to AC, it is also perpendicular to BM. Therefore, triangle OMB is a right-angled triangle.

Let's use trigonometry to find the length of AC and AB.

In triangle OMB: - The length of OM (height) is equal to the radius of the inscribed circle, which is given as 5. - The angle between OM and BM is 60°.

Using trigonometry, we can find the length of BM (base) as follows:

BM = OM * tan(angle)

Substituting the values: BM = 5 * tan(60°)

Using the value of tan(60°) = √3, we have: BM = 5 * √3

Since BM is the shorter diagonal AC, we have: AC = 2 * BM = 2 * 5 * √3 = 10 * √3

To find the length of AB (the longer diagonal), we can use the Pythagorean theorem in triangle OMB.

In triangle OMB: - The length of OM (height) is given as 5. - The length of BM (base) is given as 5 * √3.

Using the Pythagorean theorem, we can find the length of AB as follows:

AB = √(OM^2 + BM^2)

Substituting the values: AB = √(5^2 + (5 * √3)^2) = √(25 + 75) = √100 = 10

Therefore, the length of the longer diagonal AB is 10.

Answer:

The length of the longer diagonal of the rhombus is 10 units.