Question

Около првильного треугольника описана окружность и в него вписана окружность.Длина меньшей окружности равно 8пи. Найдите площадь кольца и площадь треугольника. помогите,пожалуйста)

Answer 1

r = h/3, S = r²π = 8π , r = 2√2

2√2 = h/3, h = 6√2

R = 2h/3 , R = 2*6√2/3 = 4√2.

площадь кольца найдем: S1 - S2 (S1 - площадь большей окружности с радиусом R, S2 - площадь меньшей окружности с радиусом r)

S1 = 32π

S2 = 8π, S3 = 32π-8π = 24π - площадь кольца.

площадь равностороннего треугольника равна: a²√3/4

зная высоту h = 6√2 по теореме пифагора найдем сторону а = 4√6

подставим в формулу: Sтр =  (4√6)²*√3/4 = 24√3

Answer 2

Problem Analysis

We are given a right triangle with a circumcircle and an inscribed circle. The length of the smaller circle is given as 8π. We need to find the area of the ring (the region between the two circles) and the area of the triangle.

Solution

To solve this problem, we need to use some properties of right triangles and circles.

Let's denote the radius of the smaller circle as r and the radius of the larger circle as R.

Finding the Radius of the Smaller Circle

We are given that the length of the smaller circle is 8π. The formula for the circumference of a circle is C = 2πr, where C is the circumference and r is the radius. We can set up the following equation:

8π = 2πr

Simplifying the equation, we find:

r = 4

So, the radius of the smaller circle is 4.

Finding the Radius of the Larger Circle

In a right triangle, the radius of the circumcircle is equal to half the length of the hypotenuse. In this case, the hypotenuse is the diameter of the larger circle. Let's denote the length of the hypotenuse as c.

Since the triangle is a right triangle, we can use the Pythagorean theorem to find the length of the hypotenuse:

a^2 + b^2 = c^2

where a and b are the lengths of the other two sides of the triangle.

Since the triangle is a right triangle, one of the angles is 90 degrees. Therefore, we can use the properties of right triangles to find the lengths of the sides.

Let's denote the lengths of the other two sides as a and b.

Since the triangle is a right triangle, we can use the following relationships:

a = 2r b = 2r

Substituting these values into the Pythagorean theorem, we get:

(2r)^2 + (2r)^2 = c^2

Simplifying the equation, we find:

8r^2 + 8r^2 = c^2

16r^2 = c^2

Taking the square root of both sides, we find:

c = 4√2r

So, the length of the hypotenuse (diameter of the larger circle) is 4√2r.

Since the radius of the larger circle is half the length of the hypotenuse, we have:

R = 2√2r

Substituting the value of r, we find:

R = 2√2 * 4

Simplifying the equation, we find:

R = 8√2

So, the radius of the larger circle is 8√2.

Finding the Area of the Ring

The area of the ring is equal to the difference between the areas of the two circles.

The formula for the area of a circle is A = πr^2, where A is the area and r is the radius.

The area of the smaller circle is:

A1 = π(4)^2 = 16π

The area of the larger circle is:

A2 = π(8√2)^2 = 64π(2)

The area of the ring is:

A_ring = A2 - A1 = 64π(2) - 16π = 48π

So, the area of the ring is 48π.

Finding the Area of the Triangle

The area of a right triangle can be calculated using the formula A = (1/2) * base * height.

In this case, the base and height of the triangle are equal to the lengths of the legs of the right triangle.

The length of each leg is equal to 2r, so the base and height of the triangle are both equal to 2r.

The area of the triangle is:

A_triangle = (1/2) * 2r * 2r = 2r^2 = 2(4)^2 = 32

So, the area of the triangle is 32.

Summary

To summarize, the area of the ring between the two circles is 48π, and the area of the right triangle is 32.