In a right-angled triangle ABC C = 90 degrees AB = 10 cm angle ABC = 30 degrees A

In a right-angled triangle ABC C = 90 degrees AB = 10 cm angle ABC = 30 degrees A circle is drawn with the center at point A. What should be the radius of this circle so that 1. The circle touches the straight line BC 2. Does not have common points with it. 3. Had 2 common points with it.

Find the length of the AC leg. From the properties of a right-angled triangle, it is known that opposite an angle of 30 degrees, there is a leg, which is exactly 2 times less than the hypotenuse. This property can be proved using the definition of the sine of an angle: the sine of an angle in a right-angled triangle is the ratio of the opposite leg to the hypotenuse. The sine of the angle ABC is equal to:
sinABC = AC / AB;
sin30 = AC / 10;
1/2 = AC / 10;
AC = 10 * 1/2 (in proportion);
AC = 5 cm.
Thus:
1. for the circle to touch the straight line BC, the radius must be 5 cm;
2. so that the circle does not have common points with the straight line BC, the radius should be less than 5 cm;
3. for the circle to have 2 common points with the straight line BC, the radius must be greater than 5 cm.
Answer: 1. R = 5 cm;
2. R <5 cm;
3.R> 5 cm.