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Home > GC > Chapter 11 > Lesson 11.2.3 > Problem 11-96

11-96.

Solve for the variables in each of the diagrams below. Assume point is the center of the circle in part (b).  

  1. 2 rays form an angle, outside a circle, labeled, 90 degrees. Each ray, tangent to the circle, divide the circle into 2 arcs, smaller is unlabeled, larger labeled, x.

    Added to diagram, dashed segments from center of circle to points of tangency, creating square.

  1. 2 tangents form an angle, labeled, 48 degrees, outside circle with center, c. Dashed segments from, c, to each point of tangency, labeled, 7, and central angle, labeled, x, with tangent segment, labeled, y.

    Added to diagram, right angles between radii & point of tangency, dashed line from angle vertex, to, C, and 1 tick mark on each tangent segment.

    By , the two triangles are congruent. Therefore, the corresponding angles of the two triangles are also congruent, which means the values of the angles are equal to ,  (half the value of the original ), and (half the original value of ).

    By the Law of Sines:

  1. Circle with 2 intersecting chords, with segments around point of intersection, labeled as follows: top, left side, 3, right side, x, bottom, left side, 6, right side, x, + 2.

    Added to diagram, line segments, connecting the ends of the chords, create 2 triangles.

    The two triangles are similar, which means:

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