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Geometry: Theorems

Theorems for Segments and Circles

Problems

Problems

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

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.

Figure %: Chords of the same circle that intersect

In the figure above, chords QR and ST intersect. The theorem states that the product of QB and BR is equal to the product of SB and BT.

Theorem 2

Every secant segment is divided into two segments by the circle it intersects. The internal segment is a chord, and the external segment is the segment with one endpoint at the intersection of the secant segment and the circle, and the other endpoint at the fixed point outside the circle. Given these conditions, a theorem states that when two secant segments share an endpoint not on the circle, the products of the lengths of each secant segment and its external segment are equal.

Figure %: Secant segments that share an endpoint not on the circle
In the figure above, the secant segments DE and FE share an endpoint, E, outside the circle. The theorem states that the product of the lengths of DE and ME is equal to the product of the lengths of FE and NE.

Theorem 3

A similar theorem exists when a secant segment and a tangent segment share an endpoint not on the circle. This theorem states that the length of the tangent segment squared is equal to the product of the secant segment and its external segment.

Figure %: A secant segment and a tangent segment that share an endpoint not on the circle
In the figure above, secant segment QR and tangent segment SR share an endpoint, R, not on the circle. The theorem states that the length of SR squared is equal to the product of the lengths of QR and KR.

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