Take Note Theorem 12-3

Theorem

If two tangent segments to a circle share a common endpoint outside the circle, then the two segments are congruent.

If …

B A ¯ b eh bar  and B C ¯ b c bar  are tangent to ⊙ O circle dot o

Then …

B A ¯ ≅ B C ¯ b eh bar , approximately equal to , b c bar

You will prove Theorem 12-3 in Exercise 23.

Problem 5 Circles Inscribed in Polygons

⊙ O circle dot o  is inscribed in Δ cap delta ABC. What is the perimeter of Δ cap delta ABC?

A D = A F = 10 c m B D = B E = 15 c m C F = C E = 8 c m Two segments tangent to a circle from a point outside the circle are congruent , so they have the same length . table with 1 row and 2 columns , row1 column 1 , table with 3 rows and 1 column , row1 column 1 , eh d equals eh f equals 10 c m , row2 column 1 , b d equals b e equals 15 c m , row3 column 1 , c f equals c e equals 8 c m , end table , column 2 table with 3 rows and 1 column , row1 column 1 , cap twosegmentstangenttoacirclefrom , row2 column 1 , apointoutsidethecirclearecongruent . comma , row3 column 1 , sotheyhavethesamelength . . , end table , end table

p = A B + B C + C A Definition of perimeter  p   = A D + D B + B E + E C + C F + F A Segment Addition Postulate   = 10 + 15 + 15 + 8 + 8 + 10 Substitute .   = 66   table with 4 rows and 3 columns , row1 column 1 , p , column 2 equals eh b plus b c plus c eh , column 3 cap definitionofperimeter . p , row2 column 1 , , column 2 equals eh d plus d b plus b e plus e c plus c f plus f eh , column 3 cap segmentcap additioncap postulate , row3 column 1 , , column 2 equals 10 plus 15 plus 15 plus 8 plus 8 plus 10 , column 3 cap substitute . . , row4 column 1 , , column 2 equals 66 , column 3 , end table

The perimeter is 66 cm.

✓ Got It?

5. ⊙ O circle dot o  is inscribed in Δ P Q R , cap delta p q r comma  which has a perimeter of 88 cm. What is the length of Q Y ¯ ? q y bar , question mark

✓ Lesson Check

Do you know HOW?

1. If m ∠ A = 58 , m angle . eh equals 58 comma  what is m ∠ A C B ? m angle eh c b question mark
2. If B C = 8 b c equals 8  and D C = 4 , d c equals 4 comma  what is the radius?
3. If A C = 12 eh c equals 12  and B C ¯ = 9 , b c bar . equals 9 comma  what is the radius?

Do you UNDERSTAND?

4. Vocabulary How are the phrases tangent ratio and tangent of a circle used differently?
5. Error Analysis A classmate insists that D F ¯ d f bar  is a tangent to ⊙ E . circle dot e .  Explain how to show that your classmate is wrong.