1. GEOMETRY HELP BA is tangent to C at point A. Find the value of x.. 90 + 22 + x = 180 Substitute. 112 + x = 180Simplify. x = 68 Solve. m A + m B + m C = 180 Triangle Angle-Sum Theorem Because BA is tangent to C, A must be a right angle. Use the Triangle Angle-Sum Theorem to find x.. Quick Check Tangent Lines LESSON 12-1 Additional Examples

2. GEOMETRY HELP Because opposite sides of a rectangle have the same measure, DW = 3 cm and OD = 15 cm. Because OZ is a radius of O, OZ = 3 cm.. 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. Draw OP. Then draw OD parallel to ZW to form rectangle ODWZ, as shown below. Tangent Lines LESSON 12-1 Additional Examples

3. GEOMETRY HELP OD 2 + PD 2 = OP 2 Pythagorean Theorem 15 2 + 4 2 = OP 2 Substitute. 241 = OP 2 Simplify. The distance between the centers of the pulleys is about 15.5 cm. Because the radius of P is 7 cm, PD = 7 – 3 = 4 cm.. (continued) Because ODP is the supplement of a right angle, ODP is also a right angle, and OPD is a right triangle. Quick Check Tangent Lines LESSON 12-1 Additional Examples OP Use a calculator to find the square root.

4. GEOMETRY HELP 144  194Simplify. 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..... 12 2 13 2 + 5 2 Substitute. Because PO 2  PA 2 + OA 2, PA is not tangent to O at A.. PO 2 PA 2 + OA 2 Is OAP a right triangle? 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 PO 2 = PA 2 + OA 2.. Tangent Lines LESSON 12-1 Additional Examples Quick Check

5. GEOMETRY HELP 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.. Theorem 12-3 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. Because QS and QT are tangent to O, QS QT, so QS = QT.. Tangent Lines LESSON 12-1 Additional Examples Quick Check

6. GEOMETRY HELP p = XY + YZ + ZW + WX Definition of perimeter p = XR + RY + YS + SZ + ZT + TW + WU + UX Segment Addition Postulate = 11 + 8 + 8 + 6 + 6 + 7 + 7 + 11 Substitute. = 64 Simplify. The perimeter is 64 ft. XU = XR = 11 ft YS = YR = 8 ft ZS = ZT = 6 ft WU = WT = 7 ft By Theorem 12-3, two segments tangent to a circle from a point outside the circle are congruent. C is inscribed in quadrilateral XYZW. Find the perimeter of XYZW.. Tangent Lines LESSON 12-1 Additional Examples Quick Check