1. Splash Screen

2. Then/Now You used the Pythagorean Theorem to find side lengths of right triangles. Use properties of tangents. Solve problems involving circumscribed polygons.

3. Vocabulary tangent point of tangency common tangent

4. A tangent is a line in the same plane as a circle and intersects the circle at only one point called the point of tangency

5. A common tangent is a line, ray, or segment that is tangent to two circles in the same plane

6. Example 1 Identify Common Tangents A. Copy the figure and draw the common tangents. If no common tangent exists, state no common tangent. Answer: These circles have no common tangents. Any tangent of the inner circle will intercept the outer circle in two points.

7. Example 1 Identify Common Tangents B. Copy the figure and draw the common tangents. If no common tangent exists, state no common tangent. Answer: These circles have 2 common tangents.

8. Example 1 A.2 common tangents B.3 common tangents C.4 common tangents D.no common tangents B. Copy the figure and draw the common tangents to determine how many there are. If no common tangent exists, choose no common tangent.

9. Concept

10. Example 2 Identify a Tangent Test to see if ΔKLM is a right triangle. ? 20 2 + 21 2 = 29 2 Pythagorean Theorem 841 =841 Simplify. Answer:

11. Example 2 A. B.

12. Example 3 Use a Tangent to Find Missing Values EW 2 + DW 2 =DE 2 Pythagorean Theorem 24 2 + x 2 =(x + 16) 2 EW = 24, DW = x, and DE = x + 16 576 + x 2 =x 2 + 32x + 256Multiply. 320 =32xSimplify. 10 =xDivide each side by 32. Answer: x = 10

13. Example 3 A.6 B.8 C.10 D.12

14. Concept

15. Example 4 Use Congruent Tangents to Find Measures AC =BCTangents from the same exterior point are congruent. 3x + 2 =4x – 3Substitution 2 =x – 3Subtract 3x from each side. 5 =xAdd 3 to each side. Answer: x = 5

16. Example 4 A.5 B.6 C.7 D.8

17. Example 5 Find Measures in Circumscribed Polygons Step 1Find the missing measures.

18. Example 5 Find Measures in Circumscribed Polygons Step 2Find the perimeter of ΔQRS. Answer: So, the perimeter of ΔQRS is 36 cm. = 10 + 2 + 8 + 6 + 10 or 36 cm

19. Example 5 A.42 cm B.44 cm C.48 cm D.56 cm