Take Note Theorem 12-15

For a given point and circle, the product of the lengths of the two segments from the point to the circle is constant along any line through the point and circle.

1. 

   a · b = c · d eh middle dot , b equals . c middle dot d

2. 

   ( w + x ) w = ( y + z ) y open w plus . x close w equals . open y plus z , close y

3. 

   ( y + z ) y = t 2 open y plus . z close y equals . t squared

Proof

Proof of Theorem 12-15, Case I

Given: A circle with chords A B ¯ eh b bar  and C D ¯ c d bar  intersecting at P

Prove: a · b = c · d eh middle dot , b equals . c middle dot d

Draw A C ¯ eh c bar  and B D ¯ . ∠ A ≅ ∠ D b d bar , . . angle , eh approximately equal to , angle d  and ∠ C ≅ ∠ B angle , c approximately equal to , angle b  because each pair intercepts the same arc, and angles that intercept the same arc are congruent. Δ A P C ∼ Δ D P B cap delta eh p c tilde operator cap delta d p b  by the Angle-Angle Similarity Postulate. The lengths of corresponding sides of similar triangles are proportional, so a d = c b . eh over d , equals , c over b , .  Therefore, a · b = c · d . eh middle dot , b equals . c middle dot d .

Problem 3 Finding Segment Lengths

Algebra Find the value of the variable in ⊙ N . circle dot n .

1. 

   ( 6 + 8 ) 6 = ( 7 + y ) 7 Thm . 12 - 15 ,  Case II 84 = 49 + 7 y Distributive Property 35 = 7 y   5 = y Solve for  y . table with 4 rows and 3 columns , row1 column 1 , open 6 plus 8 close 6 , column 2 equals open 7 plus y close 7 , column 3 cap thm . 12 minus 15 comma , cap casecap icap i , row2 column 1 , 84 , column 2 equals 49 plus 7 y , column 3 cap distributivecap property , row3 column 1 , 35 , column 2 equals 7 y , column 3 , row4 column 1 , 5 , column 2 equals y , column 3 cap solvefor . y . , end table

2. 

   ( 8 + 16 ) 8 = z 2 Thm . 12 - 15 ,  Case III 192 = z 2 Simplify . 13.9 ≈ z Solve for  z . table with 3 rows and 3 columns , row1 column 1 , open 8 plus 16 close 8 , column 2 equals , z squared , column 3 cap thm . 12 minus 15 comma . cap casecap icap icap i , row2 column 1 , 192 , column 2 equals , z squared , column 3 cap simplify , . , row3 column 1 , 13.9 , column 2 almost equal to z , column 3 cap solvefor . z . , end table