Question

ABCD-квадрат, AC=10 корень из 2, O-середина AD. С центром в точке O проведена окружность. Каким должен быть радиус этой окружности, чтобы:1) окружность касалась прямых AB и CD2) не имела с ними общих точек3) имела бы две общие точки с каждой прямой?

Answer 1

1) 5√2
2) меньше 5√2 или больше 10
3) 5 ≤ х ≤ 10

Answer 2

Problem Analysis

We are given a square ABCD with side length AC = 10√2 and O as the midpoint of AD. We need to determine the radius of a circle centered at point O that satisfies the following conditions: 1) The circle is tangent to the lines AB and CD. 2) The circle does not intersect the lines AB and CD. 3) The circle intersects each line at two distinct points.

Solution

To solve this problem, we can use the properties of tangents to circles. Let's analyze each condition separately.

Condition 1: The circle is tangent to the lines AB and CD

For a circle to be tangent to a line, the distance from the center of the circle to the line must be equal to the radius of the circle. In this case, the center of the circle is point O, and the radius is r (which we need to find).

Let's consider the line AB. The distance from point O to line AB is equal to the perpendicular distance from O to AB. Since AB is a horizontal line, this distance is equal to the vertical distance from O to AB. We can find this distance using the Pythagorean theorem.

Let's denote the perpendicular distance from O to AB as h. We can form a right triangle OAB, where OA = OB = AC/2 = 5√2. The hypotenuse of this triangle is the distance from O to A or B, which is AC/2 = 5√2. Using the Pythagorean theorem, we can find h:

h^2 + (AC/2)^2 = OA^2 h^2 + (5√2)^2 = (5√2)^2 h^2 + 50 = 50 h^2 = 0 h = 0

Since the perpendicular distance from O to AB is 0, the circle is tangent to line AB.

Similarly, we can show that the circle is tangent to line CD.

Therefore, the radius of the circle should be 0 to satisfy condition 1.

Condition 2: The circle does not intersect the lines AB and CD

Since the radius of the circle is 0, the circle does not intersect the lines AB and CD. Therefore, condition 2 is automatically satisfied.

Condition 3: The circle intersects each line at two distinct points

Since the radius of the circle is 0, the circle does not intersect the lines AB and CD. Therefore, condition 3 is not satisfied.

Conclusion

In order for the circle to satisfy all three conditions, the radius of the circle should be 0.

Answer: The radius of the circle should be 0.