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AB is diameter of semi-circle k, C is an arbitrary point on the semi-circle (other than A or B) and S is the centre of the circle inscribed into ∆ABC, then measures of angle ASB =?.?
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AB is diameter of semi-circle k, C is an arbitrary point on the semi-c...
Introduction:
In this problem, we are given a semi-circle with diameter AB. We need to find the measures of angle ASB, where C is an arbitrary point on the semi-circle (other than A or B) and S is the center of the circle inscribed in triangle ABC.

Approach:
To find the measure of angle ASB, we will use the properties of angles in a circle, inscribed angles, and angles in a triangle.

Key Pointers:
1. Angle in a semi-circle is always a right angle. Therefore, angle ACB is a right angle.
2. Angle ASB is an inscribed angle, and its measure is half the measure of the intercepted arc AB.
3. The center of the circle inscribed in triangle ABC, S, is the intersection point of the angle bisectors of angles ABC, BAC, and ACB. It is equidistant from the sides of the triangle.
4. The incenter S is also the intersection point of the perpendicular bisectors of the sides of triangle ABC.

Solution:

Step 1: Draw a semi-circle with diameter AB.

Step 2: Draw an arbitrary point C on the semi-circle (other than A or B).

Step 3: Draw the triangle ABC with the given points A, B, and C.

Step 4: Draw the perpendicular bisectors of the sides of triangle ABC. The intersection point is the center of the inscribed circle, S.

Step 5: Join AS and BS.

Step 6: Since ACB is a right angle (angle in a semi-circle), we have angle ACB = 90 degrees.

Step 7: Since S is the incenter of triangle ABC, we have AS = BS (S is equidistant from the sides of the triangle).

Step 8: As angle ACB = 90 degrees, angle ASB is an inscribed angle and its measure is half the measure of the intercepted arc AB.

Step 9: Therefore, angle ASB = 1/2 * measure of arc AB.

Conclusion:
In conclusion, the measure of angle ASB is equal to half the measure of the intercepted arc AB.
Community Answer
AB is diameter of semi-circle k, C is an arbitrary point on the semi-c...
C=135� for all value of c
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AB is diameter of semi-circle k, C is an arbitrary point on the semi-circle (other than A or B) and S is the centre of the circle inscribed into ∆ABC, then measures of angle ASB =?.?
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