Segment Lengths in Circles MM2G3. Students will understand the properties of circles. a. Understand and use properties of chords, tangents, and secants as an application of triangle similarity.

Theorem 6.16 If two chords intersect in the interior of a circle, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord.

EXAMPLE 1 Find ML and JK. SOLUTION NK NJ = NL NM x (x + 4) Use Theorem 6.16. 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.

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

Theorem 6.17 If two secant segments share the same endpoint outside a circle, then the product of the lengths of one secant segment and its external segment equals the product of the lengths of the other secant segment and its external segment.

EXAMPLE 2 SOLUTION RQ RP = RS RT 4 (5 + 4) = 3 (x + 3) 36 = 3x + 9 Use Theorem 6.17 4 (5 + 4) = 3 (x + 3) Substitute. 36 = 3x + 9 Simplify. 9 = x Solve for x ANSWER The correct answer is D.

GUIDED PRACTICE Find the value(s) of x. 13 = x ANSWER

GUIDED PRACTICE Find the value(s) of x. x = 8 ANSWER

GUIDED PRACTICE Find the value(s) of x.

Theorem 6.18 If a secant segment and a tangent segment share an endpoint outside a circle, then the product of the lengths of the secant segment and its external segment equals the square of the length of the tangent segment.

GUIDED PRACTICE 3. Find the value of x. x = ANSWER

GUIDED PRACTICE 4. Find the value of x. x = 24 5 ANSWER

GUIDED PRACTICE 5. Find the value of x. x = 8 ANSWER

GUIDED PRACTICE 6. Determine which theorem you would use to find x. Then find the value of x. x = 8 ANSWER

GUIDED PRACTICE 7. Find the value of x.