Question

Помогите пожалуйста)С задачей :К окружности с центром О проведена касательная АР(Р-точка касания). Найдите площадь треугольника АОР, если ∠ОАР=30°,а радиус окружности равен 4.Заранее благодарю)

Answer 1

Т.к. <OAP=30градусов, то PO=R=0,5AO=>AO=2*4см=8см. По теореме Пифагора: AP=см=см=4см. Sapo=4*4*0,5=8см^2.
Ответ: Sapo=8см^2.

Answer 2

Problem Analysis

We are given a circle with center O and a tangent line AR, where R is the point of tangency. We need to find the area of triangle AOR, given that angle OAR is 30 degrees and the radius of the circle is 4.

Solution

To find the area of triangle AOR, we can use the formula for the area of a triangle given two sides and the included angle. In this case, we know the length of side AO (which is the radius of the circle) and the length of side AR (which is equal to the radius of the circle, since AR is a tangent line).

Let's denote the length of side AO as a and the length of side AR as b. Since AO and AR are both radii of the circle, we have a = b = 4.

The formula for the area of a triangle given two sides and the included angle is:

Area = (1/2) * a * b * sin(angle)

Substituting the values we know:

Area = (1/2) * 4 * 4 * sin(30°)

Now we can calculate the area using this formula.

Calculation

Using the given formula, we can calculate the area of triangle AOR:

Area = (1/2) * 4 * 4 * sin(30°)

Calculating the value of sin(30°):

Area = (1/2) * 4 * 4 * 0.5

Area = 4 * 2

Area = 8 square units

Therefore, the area of triangle AOR is 8 square units.

I hope this helps! Let me know if you have any further questions.