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Geometry 9.5 Tangents to Circles

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Objectives Identify segments and lines related to circles. Use properties of a tangent to a circle.

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Some definitions you need Circle set of all points in a plane that are equidistant from a given point called a center of the circle. A circle with center P is called circle P, or P. The distance from the center to a point on the circle is called the radius of the circle. Two circles are congruent if they have the same radius.

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Some definitions you need The distance across the circle, through its center is the diameter of the circle. The diameter is twice the radius. The terms radius and diameter describe segments as well as measures.

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Some definitions you need A radius is a segment whose endpoints are the center of the circle and a point on the circle. QP, QR, and QS are radii of Q. All radii of a circle are congruent.

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Some definitions you need A chord is a segment whose endpoints are points on the circle. PS and PR are chords. A diameter is a chord that passes through the center of the circle. PR is a diameter.

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Some definitions you need A secant is a line that intersects a circle in two points. Line k is a secant. A tangent is a line in the plane of a circle that intersects the circle in exactly one point. Line j is a tangent.

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Identifying Special Segments and Lines Tell whether the line or segment is best described as a chord, a secant, a tangent, a diameter, or a radius of C. a.AD b.CD c.EG d.HB

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More information you need-- In a plane, two circles can intersect in two points, one point, or no points. Coplanar circles that intersect in one point are called tangent circles. Coplanar circles that have a common center are called concentric. 2 points of intersection.

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Tangent circles A line or segment that is tangent to two coplanar circles is called a common tangent. A common internal tangent intersects the segment that joins the centers of the two circles. A common external tangent does not intersect the segment that joins the center of the two circles. Internally tangent Externally tangent

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Concentric circles Circles that have a common center are called concentric circles. Concentric circles No points of intersection

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Identifying common tangents Tell whether the common tangents are internal or external.

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Verifying a Tangent to a Circle Determine if EF is tangent to circle D. Because 11 2 + 60 2 = 61 2 DEF is a right triangle and DE is perpendicular to EF. EF is tangent to circle D.

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Find Radius (r + 8) 2 = r 2 + 16 2 Pythagorean Thm. Substitute values c 2 = a 2 + b 2 r 2 + 16r + 64 = r 2 + 256Square of binomial 16r + 64 = 256 16r = 192 r = 12 Subtract r 2 from each side. Subtract 64 from each side. Divide. The radius of the circle is 12 feet.

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Using properties of tangents AB is tangent to C at B. AD is tangent to C at D. Find the value of x. x 2 + 2

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11 = x 2 + 2 Two tangent segments from the same point are Substitute values AB = AD 9 = x 2 Subtract 2 from each side. 3 = xFind the square root of 9. The value of x is 3 or -3. AB is tangent to C at B. AD is tangent to C at D. Find the value of x. x 2 + 2

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Practice

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X = 400+ 225 x = 625 X = 25

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40 + 30 = 2500 Hypotenuse = 2500 = 50 X = 50-30 = 20 Hypotenuse = 21 + 8 = 29 29 - 21 = 400 X = 400 = 20

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AC = BC 40 = 3x + 4 36 = 3x X = 9 Pythagorean Formula BP = AB + AP (8 + x) = x + 12 x + 16x + 64 = x + 144 16x = 80 X = 5

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CD = ED = 4 BC = AB = 3 X = BC + CD X = 3+4 = 7 Trigonometric Formula Cos 30 = 6 / x x = 6 / Cos 30 x = 6.9 30

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Pythagorean Formula BP = CP + CB X = 12 - 4 x = 144-16 X = 11.3 PB = 6 PC = 6 + 4 = 10 PQ = 6 Pythagorean Formula QC = PC -PQ QC = 100-36 = 64 QC = 8 Pythagorean Formula CD = CQ - QD CD = 64 40.96 CD = 4.8