Objectives: Use relationships between a radius and a tangent

1. Section 11-1 Tangent Lines SPI 32B: Identify chords of circles given a diagramSPI 33A: Solve problems involving the properties of arcs, tangents, chords • Objectives: • Use relationships between a radius and a tangent Tangent to a Circle Line in the plane of a circle that intersects the circle in exactly 1 point (Line AB is tangent to the circle) A B Point of Tangency Point where circle and a tangent intersect (B is the point of Tangency)

2. Relate Tangent and Radius of a Circle The Theorem can be used to solve problems.

3. Do Now . . Because BA is tangent to C, A must be a right angle. Use the Triangle Angle-Sum Theorem to find x. m A + m B + m C = 180 Triangle Angle-Sum Theorem Tangent Lines BA is tangent to C at point A. Find the value of x. 90 + 22 + x = 180 Substitute. 112 + x = 180 Simplify. x = 68 Solve.

4. Do Now . Draw OP. Then draw OD parallel to ZW to form rectangle ODWZ, as shown below. Because OZ is a radius of O, OZ = 3 cm. Real World and Tangent Lines A belt fits tightly around two circular pulleys, as shown below. Find the distance between the centers of the pulleys. Round your answer to the nearest tenth. Because opposite sides of a rectangle have the same measure, DW = 3 cm and OD = 15 cm.

5. . Because ODP is the supplement of a right angle, ODP is also a right angle, and OPD is a right triangle. Because the radius of P is 7 cm, PD = 7 – 3 = 4 cm. OP 15.524175 Use a calculator to find the square root. (continued) OD2 + PD2 = OP2Pythagorean Theorem 152 + 42 = OP2Substitute. 241 = OP2Simplify. The distance between the centers of the pulleys is about 15.5 cm.

6. Finding a Tangent Converse of Theorem 11-1

7. Do Now . . . . . . Draw the situation described in the problem. For PA to be tangent to O at A, A must be a right angle, OAP must be a right triangle, and PO2 = PA2 + OA2. PO2PA2 + OA2Is OAP a right triangle? 122 132 + 52Substitute. / 144 = 194 Simplify. / Because PO2 = PA2 + OA2, PA is not tangent to O at A. Tangent Lines O has radius 5. Point P is outside O such that PO = 12, and point A is on O such that PA = 13. Is PA tangent to O at A? Explain.

8. Using Multiple Tangents Recall When a circle is inscribed in a triangle, the triangle is circumscribed about the circle. What is the relationship between each side of the triangle and the circle? Each segment is tangent to the circle, meaning each line is perpendicular to the radius forming a right angle.

9. Using Tangents to Solve Problems

10. Do Now . . QS and QT are tangent to O at points S and T, respectively. Give a convincing argument why the diagonals of quadrilateral QSOT are perpendicular. Because QS and QT are tangent to O, QS QT, so QS = QT. Using Theorem Theorem 11-3 states that two segments tangent to a circle from a point outside the circle are congruent. OS = OT because all radii of a circle are congruent. Two pairs of adjacent sides are congruent. Quadrilateral QSOT is a kite if no opposite sides are congruent or a rhombus if all sides are congruent. By theorems in Lessons 6-4 and 6-5, both the diagonals of a rhombus and the diagonals of a kite are perpendicular.

11. Do Now . C is inscribed in quadrilateral XYZW. Find the perimeter of XYZW. XU = XR = 11 ft YS = YR = 8 ft ZS = ZT = 6 ft WU = WT = 7 ft By Theorem 11-3, two segments tangent to a circle from a point outside the circle are congruent. Tangent Lines p = XY + YZ + ZW + WXDefinition of perimeter p = XR + RY + YS + SZ + ZT + TW + WU + UXSegment Addition Postulate = 11 + 8 + 8 + 6 + 6 + 7 + 7 + 11 Substitute. = 64 Simplify. The perimeter is 64 ft.