9.5 Tangents to Circles Geometry

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

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.

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.

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.

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.

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.

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. AD CD EG HB

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.

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

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

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

Verifying a Tangent to a Circle Determine if EF is tangent to circle D. Because 112 + 602 = 612 ∆DEF is a right triangle and DE is perpendicular to EF. EF is tangent to circle D.

The radius of the circle is 12 feet. Find Radius c2 = a2 + b2 Pythagorean Thm. (r + 8)2 = r2 + 162 Substitute values r 2 + 16r + 64 = r2 + 256 Square of binomial Subtract r2 from each side. 16r + 64 = 256 Subtract 64 from each side. 16r = 192 r = 12 Divide. The radius of the circle is 12 feet.

Using properties of tangents AB is tangent to C at B. AD is tangent to C at D. Find the value of x. x2 + 2

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. x2 + 2 x2 + 2 AB = AD Two tangent segments from the same point are  11 = x2 + 2 Substitute values 9 = x2 Subtract 2 from each side. 3 = x Find the square root of 9. The value of x is 3 or -3.

Practice

X ²= 400+ 225 x ² = 625 X = 25

40² + 30² = 2500 Hypotenuse = √2500 = 50 X = 50-30 = 20 Hypotenuse = 21 + 8 = 29 29² - 21 ² = 400 X = √400 = 20

Pythagorean Formula BP² = AB² + AP² (8 + x) ² = x ² + 12² x² + 16x + 64 = x² + 144 16x = 80 X = 5 AC = BC 40 = 3x + 4 36 = 3x X = 9

30° 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

PB = 6 PC = 6 + 4 = 10 PQ = 6 Pythagorean Formula QC² = PC ² -PQ ² QC² = 100-36 = 64 QC = 8 CD² = CQ ² - QD ² CD ² = 64 – 40.96 CD = 4.8 Pythagorean Formula BP² = CP² + CB² X ² = 12 ² - 4² x² = 144-16 X = 11.3