Answer :
To solve this question, we need to evaluate the truthfulness of each given statement about properties of circles:
A. The measure of an inscribed angle is half the measure of the arc which subtended it.
This is a true property. By definition, an inscribed angle is formed by two chords of a circle that meet at a point on the circle. The inscribed angle theorem states that the measure of an inscribed angle is always half the measure of the central angle that subtends the same arc. Therefore, statement A is true.
B. The perpendicular bisector of a chord passes through the center of the circle.
This is also a true property. A chord of a circle is a line segment with both endpoints on the circle. The perpendicular bisector of a chord is a line that is perpendicular to the chord and divides it into two equal segments. According to the perpendicular bisector theorem, the perpendicular bisector of a chord of a circle passes through the center of the circle. Statement B is true.
C. An inscribed angle subtended by a semi-circle is a right angle.
This statement refers to Thales' theorem, a well-known property of circles. Thales' theorem states that if an inscribed angle intercepts a diameter of the circle (which forms a semi-circle), then the angle is a right angle (90 degrees). Statement C is true.
D. An inscribed angle subtended by a minor arc is an obtuse angle.
This statement is not always true. A minor arc is an arc of a circle that measures less than 180 degrees. The inscribed angle subtending a minor arc can be acute (less than 90 degrees), right (exactly 90 degrees), or obtuse (more than 90 degrees), depending on the length of the arc. For instance, if the minor arc is very small (close to 0 degrees), the inscribed angle will be very small (close to 0 degrees) as well, making it an acute angle, not obtuse. Therefore, statement D is not true because it implies all inscribed angles subtended by a minor arc are obtuse, which is incorrect.
The correct answer is:
D. An inscribed angle subtended by a minor arc is an obtuse angle.
This is the statement that is NOT TRUE among the given options.
