Question

к окружности с центром в точке О проведены : касательная АВ и секущая АО.Найдите радиус окружности,если АВ равно 12 см АО равен 13 см.Распишите!!!

Answer 1

смотри. секущая проведена через центр окружности. касательная касается под углом - 90 градусов соединяешь О и В и получается прямоугольный треугольник. далее по теореме пифагора АВ= корень из 169-144 и получается корень из 25 а это 5 вот и все решение)

Answer 2

Problem Statement

Дана окружность с центром в точке O. Проведены касательная AB и секущая AO. Найдите радиус окружности, если AB равно 12 см и AO равен 13 см.

Solution

To find the radius of the circle, we can use the properties of tangents and secants.

Let's analyze the given information: - AB is a tangent to the circle. - AO is a secant of the circle.

Step 1: Let's draw a diagram to visualize the problem:

          A
          |
          |
          |   O
          |
          |
          B

Step 2: Recall the properties of tangents and secants: - A tangent to a circle is perpendicular to the radius drawn to the point of tangency. - A secant is a line that intersects a circle in two distinct points.

Step 3: Apply the properties of tangents and secants to the given information: - Since AB is a tangent to the circle, it is perpendicular to the radius drawn to the point of tangency (point A). - AO is a secant of the circle, intersecting the circle at points A and another point (let's call it P).

          A
          |
          |   P
          |   |
          |   |
          |   O
          |
          B

Step 4: Use the given information to solve for the radius of the circle: - We know that AO = 13 cm and AB = 12 cm. - From the properties of tangents and secants, we can conclude that AP is also equal to 12 cm.

          A
          |
          |   P
          |   |
          |   |
          |   O
          |
          B

Step 5: Apply the Pythagorean theorem to find the length of OP: - In triangle AOP, we have AO = 13 cm, AP = 12 cm, and OP as the unknown. - Using the Pythagorean theorem, we can write: AO^2 = AP^2 + OP^2.

          A
          |
          |   P
          |   |
          |   |
          |   O
          |
          B

Step 6: Solve for OP: - Substituting the known values, we have: 13^2 = 12^2 + OP^2. - Simplifying the equation, we get: 169 = 144 + OP^2. - Subtracting 144 from both sides, we have: OP^2 = 169 - 144 = 25. - Taking the square root of both sides, we find: OP = 5 cm.

Step 7: Determine the radius of the circle: - The radius of the circle is the distance from the center (O) to any point on the circle. - In this case, OP represents the radius of the circle. - Therefore, the radius of the circle is 5 cm.

Answer

The radius of the circle is 5 cm.