Geometry Unit 5: Circles Part 1 Chords, Secants, and Tangents

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1 Geometry Unit 5: Circles Part 1 Chords, Secants, and Tangents Name Chords and Circles: A chord is a segment that joins two points of the circle. A diameter is a chord that contains the center of the circle. Circle Theorems Theorem 1: In a circle, a radius perpendicular to a chord bisects the chord. Theorem 2: In a circle, a radius that bisects a chord is perpendicular to the chord. Theorem 3: In a circle, the perpendicular bisector of a chord passes through the center of the circle. Fill in the Reasons of the proof of Theorem 1 below Statement Reason E is the midpoint of

2 Chords and Circles: continued Theorem1 In a circle, or congruent circles, congruent chords are equidistant from the center. (converse) In a circle, or congruent circles, chords equidistant from the center are congruent. Theorem 2 In a circle, or congruent circles, congruent chords have congruent arcs. (converse) In a circle, or congruent circles, congruent arcs have congruent chords. Theorem 3 In a circle, parallel chords intercept congruent arcs. Intersecting Chords Theorem If two chords intersect in a circle, the product of the lengths of the segments of one chord equal the product of the lengths of the segments of the other. Theorem Proof: Intersecting Chords Rule: (segment piece) (segment piece) = (segment piece) (segment piece) Statement This proof uses similar triangles. Similar triangles are the same shape but different sizes. All corresponding sides in similar triangle are in the same ratio. This means sides always divide to the same result. Or that the sides in one triangle can be multiplied by one number to get the sides of second similar triangle. Reasons

3 Practice 1 : Chords and Circles 1) Find OF. 7) Given: Circle O CD = 16 AB = 16 OB = 10 marked perpendiculars Given: Circle O, Find CD. 2) 8) Given: Circle O, diameter, marked perpendicular Given: Circle O, 3) Given: Circle O, 9) marked perpendiculars, hash marks indicating congruent 4) Given: Circle O, marked perpendiculars 10) 5) Find AB. Given: Circle O, marked perpendicular 6) 11) A plane intersects a sphere. The diameter of the intersection is 8. The diameter of the sphere is 20. Given: Circle with indicated center, marked parallels Practice 1 answers(scrambled)

4 Tangents and Circles A tangent to a circle is a line in the plane of the circle that intersects the circle in exactly one point. Theorem: If a line is tangent to a circle, it is perpendicular to the radius drawn to the point of tangency. Theorem: Tangent segments to a circle from the same external point are congruent. (You may think of this as the "Hat" Theorem because the diagram looks like a circle wearing a pointed hat.) This theorem can be proven using congruent triangles and the previous theorem. The triangles shown are congruent by the Hypotenuse Leg Congruency Theorem for Right Triangles. The radii (legs) are congruent and hypotenuse is shared by both triangles. By using Corresponding Parts of Congruent Triangles are Congruent, this theorem is proven true. Common Tangents: Common tangents are lines or segments that are tangent to more than one circle at the same time.

5 Practice 2: Tangents and Circles 1) 8) 2) a = OA; b = OB; c = CB Find a, b, c. 3) 4) 9) 5) 10) 6) 7) The segment through point B is tangent to circle A. Practice 2 answers #1-9 (scrambled) YES NO 5 20 What is the equation of the tangent?

6 Rules for Chords, Secants, and Tangents Secant: In geometry, a secant line of a curve is a line that intersects two points on the curve. [ Theorem (from page 1): Intersecting Chords Theorem If two chords intersect in a circle, the product of the lengths of the segments of one chord equal the product of the lengths of the segments of the other. Intersecting Chords Rule: (segment piece) (segment piece) = (segment piece) (segment piece) Theorem: If two secant segments are drawn to a circle from the same external point, the product of the length of one secant segment and its external part is equal to the product of the length of the other secant segment and its external part. Secant-Secant Rule: (whole secant) (external part) = (whole secant) (external part) Theorem: If a secant segment and tangent segment are drawn to a circle from the same external point, the product of the length of the secant segment and its external part equals the square of the length of the tangent segment. Secant-Tangent Rule: (whole secant) (external part) = (tangent) 2 This theorem can also be stated as "the tangent being the mean proportional between the whole secant and its external part" (which yields the same final rule:

7 Practice 3: Chords, Secants and Tangents 1) 6) two chords as marked. tangent and secant. 2) 7) diameter perpendicular to chord. two chords as marked. 3) 8) two secants tangent and secant. 4) 9) two secants tangent, secant and chord. 5) 10) two secants two tangent sand secant. Practice 3 answers #1-10 (scrambled) Practice Problems: Chords, Secants and Tangents (continued)

8 11) secants and chords such that LM = 7 MN = 8 TN = 10 PR = 3 RM = 6 LT = 11 Find: PT, PN, LR, RT 12) In a circle, diameter is extended through B to an external point P. Tangent is drawn to point C on the circle. If the radius of the circle is 15, and BP = 2, find PC. 13) In circle O, diameter is perpendicular to chord at E. If CD = 20, BE = 2 and AE = 4x - 2, find the length of the diameter 14) In the accompanying diagram, and are tangent to the indicated circle. From his pasture gate, a horse sees the barn and the farm pond. The angle formed between his line of sight of the barn and his line of sight of the pond is 30 degrees. How many degrees are in angle HAY?

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