Exercise 29 Page 682

We want to prove that when two lines that are tangent to the same circle intersect, the lengths between the point of intersection and the points of tangency are equal. We can also represent this using the given diagram.

We know that AB is tangent to ⊙ P at B and AC is tangent to ⊙ P at C, and want to show that AB = AC. Examining the diagram, we notice that AB and AC make up a side of △ ABP and △ ACP, respectively.

Recall the Tangent to Circle Theorem.

Tangent to Circle Theorem
A line is tangent to a circle if and only if the line is perpendicular to a radius of the circle.

Using this theorem, we know that ∠ B and ∠ C are both right angles. Also, since BP and CP are both radii of the same circle, we know that these sides are congruent.

Since the right triangle's hypotenuse is a common side between the triangles, these sides must be congruent by the Reflexive Property of Congruence.

Now we can prove congruence between the triangles by the HL (Hypotenuse-Leg) Congruence Theorem. △ ABP ≅ △ ACP Since AB and AC are corresponding sides in our triangles, we know that AB=AC.

Let's show this as a two-column proof.

Statement	Reason
1. AB and AC are tangent to ⊙ P at B and C. Both PB and PC are radii of ⊙ P.	1. Given
2. AB⊥PB, AC⊥PC	2. Tangent to Circle Theorem
3. ∠ B and ∠ C are right angles	3. Definition of perpendicular lines.
4. PB≅ PC	4. Definition of a circle's radius
5. AP≅ AP	5. Reflexive Property of Congruence
6. △ ABP ≅ △ ACP	6. HL Congruence Theorem
7. AB ≅ AC	7. Corresponding parts are congruent

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