Question

Из точки взятой вне окружности проведены к ней две касательной. Докозать что длины касательных равны между собой.

Answer 1

Решение в скане.........

Answer 2

Proof that the lengths of the tangents drawn from an external point to a circle are equal

To prove that the lengths of the tangents drawn from an external point to a circle are equal, we can use the following approach:

Let's consider a circle with center O and radius r. Let P be an external point to the circle. From point P, we draw two tangents to the circle, touching the circle at points A and B, respectively.

To prove that the lengths of the tangents PA and PB are equal, we can use the theorem that states that tangents drawn from an external point to a circle are equal in length.

Proof:

1. Draw radii OP, OA, and OB. 2. Since OA and OB are radii of the same circle, they are equal in length: OA = OB. 3. The line segment OP is common to both triangles OPA and OPB. 4. Since OP is a radius of the circle, it is perpendicular to the tangent lines PA and PB at points A and B, respectively. 5. By the definition of a tangent, the tangent line is perpendicular to the radius at the point of contact. 6. Therefore, angles OPA and OPB are right angles. 7. By the definition of a right triangle, triangles OPA and OPB are right triangles. 8. In a right triangle, the hypotenuse is always the longest side. 9. Therefore, OP is the longest side in both triangles OPA and OPB. 10. Since OP is the longest side in both triangles, the other two sides, PA and PB, must be equal in length. 11. Hence, the lengths of the tangents PA and PB are equal.

Therefore, we have proved that the lengths of the tangents drawn from an external point to a circle are equal.

Note: The proof provided above is a general proof for the equality of tangent lengths drawn from an external point to a circle. It does not depend on specific measurements or values.