Question

Диагонали ромба 12 и 9. Найти отношение площади вписанного круга к площади ромба?

Answer 1

Сторона ромба а=√(D²+d²)/2=√(144+81)/2=√225/2=15/2=7,5
Радиус вписанного круга R=Dd/4a=12*9/4*7,5=3,6
Площадь ромба Sр=Dd/2=12*9/2=54
Площадь круга Sк=πR²=π*3,6²=12,96π
Отношение Sк/Sр=12,96π/54=6π/25=0,24π

Answer 2

Calculation of the Diagonals of the Rhombus

To find the ratio of the area of the inscribed circle to the area of the rhombus, we first need to calculate the diagonals of the rhombus with lengths 12 and 9.

The diagonals of a rhombus are perpendicular bisectors of each other and divide the rhombus into four congruent right triangles. The lengths of the diagonals can be calculated using the Pythagorean theorem.

Let's denote the lengths of the diagonals as d1 and d2.

Using the Pythagorean theorem, we can calculate the length of the diagonals as follows:

d1 = √(12^2 + 9^2) = √(144 + 81) = √225 = 15.

d2 = √(12^2 - 9^2) = √(144 - 81) = √63 ≈ 7.94.

Therefore, the lengths of the diagonals of the rhombus are approximately 15 and 7.94.

Calculation of the Area of the Rhombus

The area of a rhombus can be calculated using the formula: Area = (1/2) * d1 * d2, where d1 and d2 are the lengths of the diagonals.

Using the given lengths of the diagonals, we can calculate the area of the rhombus as follows:

Area = (1/2) * 15 * 7.94 ≈ 59.55 square units.

Therefore, the area of the rhombus is approximately 59.55 square units.

Calculation of the Area of the Inscribed Circle

The area of a circle can be calculated using the formula: Area = π * r^2, where r is the radius of the circle.

In the case of an inscribed circle, the radius is equal to half the length of the diagonal of the rhombus.

Using the length of the diagonal d1 (15), we can calculate the radius of the inscribed circle as follows:

Radius = d1/2 = 15/2 = 7.5.

Therefore, the radius of the inscribed circle is 7.5 units.

Using the radius, we can calculate the area of the inscribed circle as follows:

Area = π * (7.5)^2 = 56.25π square units.

Calculation of the Ratio of the Area of the Inscribed Circle to the Area of the Rhombus

To find the ratio of the area of the inscribed circle to the area of the rhombus, we divide the area of the inscribed circle by the area of the rhombus.

Ratio = (Area of Inscribed Circle) / (Area of Rhombus) = (56.25π) / 59.55 ≈ 2.99.

Therefore, the ratio of the area of the inscribed circle to the area of the rhombus is approximately 2.99.

Please note that the values provided are approximate due to rounding.

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