9.5 Tangents to Circles

1. 9.5 Tangents to Circles Geometry

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

3. 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.

4. 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.

5. 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.

6. 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.

7. Some definitions you need • A secant is a line that intersects a circle in two points. Linek is a secant. • A tangent is a line in the plane of a circle that intersects the circle in exactly one point. Linej is a tangent.

8. 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

9. 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.

10. 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

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

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

15. 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.

16. 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.

17. 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

18. x2 + 2 • AB is tangent to C at B. • AD is tangent to C at D. • Find the value of x. 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.

19. Practice

20. X ²= 400+ 225 x ² = 625 X = 25

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

22. 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

23. 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

24. PB = 6 PC = 6 + 4 = 10 PQ = 6 Pythagorean Formula QC² = PC ² -PQ ² QC² = 100-36 = 64 QC = 8 Pythagorean Formula CD² = CQ ² - QD ² CD ² = 64 – 40.96 CD = 4.8 Pythagorean Formula BP² = CP² + CB² X² = 12 ² - 4² x² = 144-16 X = 11.3