Question

1. Explain the difference between circumscribed angles and inscribed angles in relationship to a circle.

Answer 1

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Explanation:

1. Consider the attachment. Angle CDF is a circumscribed angle. It is formed at the intersection of two tangents to the circle. Angles BCE, CBE, and BEC are inscribed angles. Their vertices are on the circle, and they are formed by two chords.

If a chord degenerates to zero length, the ray it represents with respect to an inscribed angle becomes a tangent. That is, angles BCD and ECD can also be considered to be inscribed angles. Similarly, straight angle DFG at tangent point F can be considered to be a degenerate case of both an inscribed angle and a circumscribed angle. (Some authors may not include degenerate cases in their definitions. YMMV)

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2. The general form of the equation of a circle is ...

  Ax² +Ay² +Dx +Ey +F = 0

Any equation of a circle can be put in this form. If we divide by A, this can be written as ...

  x² +y² -2hx -2ky +c = 0 . . . . where h = -D/(2A), k = -E/(2A), c = F/A

and the equation can be further transformed to the standard form ...

  (x -h)² +(y -k)² = r² . . . . . where r² = h² +k² -c

The last equation is the result of "completing the square" by adding h² and k² to both sides of the second equation shown. The values of h and k are as shown above, given the coefficients of the general form equation.

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3. Again refer to the attached diagram.

a) a radius is a line segment with the circle center at one end and a point on the circle at the other end. AB and AC are radii.

b) A diameter is a line segment whose ends are on the circle and whose midpoint is the center of the circle. BC is a diameter.

c) A chord is a line segment whose end points lie on the circle. BC, BE, and CE are chords.

d) A secant is a line containing two points of the circle. Lines CG and CE are secants.

e) A tangent is a degenerate case of a secant in which the two points of intersection merge to one point of intersection. It is a line that touches the circle at exactly one point, the point of tangency. DG and DC are tangents.

f) The point of tangency is the single point where a tangent intersects a circle. A radius to the point of tangency (such as AC) is always perpendicular to the tangent line (DC).