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Geometry: Theorems

Theorems for Segments and Circles


Theorems for Segments and Circles, page 2

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Tangent Segments

Given a point outside a circle, two lines can be drawn through that point that are tangent to the circle. The tangent segments whose endpoints are the points of tangency and the fixed point outside the circle are equal. In other words, tangent segments drawn to the same circle from the same point (there are two for every circle) are equal.

Figure %: Tangent segments that share an endpoint not on the circle are equal


Chords within a circle can be related many ways. Parallel chords in the same circle always cut congruent arcs. That is, the arcs whose endpoints include one endpoint from each chord have equal measures.

Figure %: Arcs AC and BD have equal measures

When congruent chords are in the same circle, they are equidistant from the center.

Figure %: Congruent chords in the same circle are equidistant from the center
In the figure above, chords WX and YZ are congruent. Therefore, their distances from the center, the lengths of segments LC and MC, are equal.

A final word on chords: Chords of the same length in the same circle cut congruent arcs. That is, if the endpoints of one chord are the endpoints of one arc, then the two arcs defined by the two congruent chords in the same circle are congruent.

Intersecting Chords, Tangents, and Secants

A number of interesting theorems arise from the relationships between chords, secant segments, and tangent segments that intersect. First of all, we must define a secant segment. A secant segment is a segment with one endpoint on a circle, one endpoint outside the circle, and one point between these points that intersects the circle. Three theorems exist concerning the above segments.

Theorem 1

PARGRAPH When two chords of the same circle intersect, each chord is divided into two segments by the other chord. The product of the segments of one chord is equal to the product of the segments of the other chord.