Question

Из точки А вне круга проведена к окружности касательная AP. Вычислите расстояние от точки А до центра окружности , если радиус круга равен 3 м и отрезок касательной AP равен 4 м . Придумайте какую-нибудь задачу из повседневной жизни, которую можно решить по этим данным.

Answer 1

Можно решить по теореме Пифагора c²=a²+b² т.е. квадрат гипотенузы равен сумме квадратов катитов Мы знаем точно, что у нас есть прямоугольный треугольник, т.к. один угол треугольника = 90° (свойства касательной) и считаем c²= 3²+4²=9+16=25, c=5 Ответ : расстояние от точки А до центра = 5м

Answer 2

Calculation of the Distance from Point A to the Center of the Circle

To calculate the distance from point A to the center of the circle, we can use the properties of tangents and the Pythagorean theorem.

Given: - Radius of the circle (r) = 3 m - Length of the tangent AP = 4 m

Let's denote the center of the circle as O and the point where the tangent AP intersects the circle as P.

To find the distance from point A to the center of the circle, we need to find the length of the line segment AO.

Since AP is tangent to the circle, it is perpendicular to the radius OP at point P. Therefore, triangle AOP is a right triangle.

Using the Pythagorean theorem, we can write:

AO^2 = AP^2 + OP^2

Substituting the given values:

AO^2 = 4^2 + 3^2

AO^2 = 16 + 9

AO^2 = 25

Taking the square root of both sides:

AO = 5 m

Therefore, the distance from point A to the center of the circle is 5 meters.

Example Problem from Everyday Life

Let's consider a scenario where you are planning to install a fence around a circular garden. The radius of the garden is 3 meters, and you want to place a gate at a point on the circumference of the garden.

You have a tangent path leading to the gate, and the length of the tangent path is 4 meters. You want to know the distance from the gate to the center of the garden to ensure proper placement of the gate.

By using the calculations we performed earlier, we can determine that the distance from the gate (point A) to the center of the garden is 5 meters.

Knowing this distance, you can accurately position the gate at a distance of 5 meters from the center of the garden, ensuring a balanced and aesthetically pleasing layout.

This information can be helpful in various situations, such as landscaping, architecture, or urban planning, where precise measurements and placement are essential.