1. Find lengths using Theorem 10.14
EXAMPLE 1
Find lengths using Theorem 10.14
Find ML and JK.
SOLUTION
NK NJ
= NL NM
Use Theorem
x (x + 4)
= (x + 1) (x + 2)
Substitute.
x2 + 4x
= x2 + 3x + 2
Simplify. 4x
= 3x + 2
Subtract x2 from each side.
x = 2
Solve for x.

2. EXAMPLE 1
Find lengths using Theorem 10.14
Find ML and JK by substitution.
SOLUTION ML
= ( x + 2 ) + ( x + 1) JK
= x + ( x + 4) = =
= 7
= 8

3. Standardized Test Practice
EXAMPLE 2
Standardized Test Practice
SOLUTION
RQ RP
= RS RT
Use Theorem
4 (5 + 4)
= (x + 3)
Substitute. 36
= 3x + 9
Simplify.
9 = x
Solve for x
ANSWER
The correct answer is D.

4. GUIDED PRACTICE
for Examples 1 and 2
Find the value(s) of x.
13 = x
ANSWER

5. GUIDED PRACTICE
for Examples 1 and 2
Find the value(s) of x.
x = 8
ANSWER

6. GUIDED PRACTICE
for Examples 1 and 2
Find the value(s) of x.
3 = x
ANSWER

7. Find lengths using Theorem 10.16
EXAMPLE 3
Find lengths using Theorem 10.16
Use the figure at the right to find RS.
SOLUTION
RQ2
= RS RT
Use Theorem
162
= x (x + 8)
Substitute.
256
= x2 + 8x
Simplify.
= x2 + 8x – 256
Write in standard form. x
–8 +
82 – 4(1) (– 256)
2(1) =
Use quadratic formula. x
= – 4 + 4 17
Simplify.

8. EXAMPLE 3
Find lengths using Theorem 10.16
Use the positive solution, because lengths cannot be negative.
So, x , and RS
= – 4 + 4 17

9. GUIDED PRACTICE
for Example 3 4.
Find the value of x. x
= 2
ANSWER

10. GUIDED PRACTICE
for Example 3 5.
Find the value of x. x = 24 5
ANSWER

11. GUIDED PRACTICE
for Example 3 6.
Find the value of x. x
= 8
ANSWER

12. GUIDED PRACTICE
for Example 3 7.
Determine which theorem you would use to find x. Then
find the value of x. x
= – 7 +
274
ANSWER

13. GUIDED PRACTICE
for Example 3 8.
Determine which theorem you would use to find x. Then
find the value of x. x
= 8
ANSWER

14. GUIDED PRACTICE
for Example 3 8.
Determine which theorem you would use to find x. Then
find the value of x. x
= 16
ANSWER

15. GUIDED PRACTICE
for Example 3 9.
In the diagram for Theorem 10.16, what must be true
about EC compared
to EA ?
EC < EA
ANSWER

16. EXAMPLE 4
Solve a real-world problem
SCIENCE
Tethys, Calypso, and Telesto are three of Saturn’s moons. Each has a nearly circular orbit 295,000 kilometers in radius. The Cassini-Huygens spacecraft entered Saturn’s orbit in July Telesto is on a point of tangency. Find the distance DB from Cassini to Tethys.

17. Solve a real-world problem
EXAMPLE 4
Solve a real-world problem
SOLUTION
DC DB
= AD2
Use Theorem
83,000 DB
203,0002
Substitute. DB
496,494
Solve for DB.
ANSWER
Cassini is about 500,000 kilometers from Tethys.

18. GUIDED PRACTICE
for Examples 4
11. Why is it appropriate to use the approximation symbol in the last two steps of the solution to Example 4?
Since the distances given are not exact, it is appropriate to use the approximation symbol in the solution.
ANSWER

19. Daily Homework Quiz 1.
Find the value of x. Round to the nearest tenth,if necessary.
ANSWER 2

20. Daily Homework Quiz
Find the value of x. Round to the nearest tenth,if necessary. 2.
ANSWER 9

21. Daily Homework Quiz
Find the value of x. Round to the nearest tenth,if necessary. 3.
ANSWER 15

22. Daily Homework Quiz
Find the value of x. Round to the nearest tenth,if necessary. 4.
ANSWER
20.25

23. Daily Homework Quiz
Find the value of x. Round to the nearest tenth,if necessary. 5.
ANSWER
11.3