Properties of Tangents Section 11.2 Properties of Tangents

Goal Use properties of a tangent to a circle.

Key Vocabulary Tangent segment

Theorems 11.1 Perpendicular Tangent Theorem 11.2 Perpendicular Tangent Converse 11.3 Congruent Tangent Segments

Tangents A tangent is a line in the plane of a circle that intersects the circle in exactly one point. Line j is a tangent. j

Theorem 11.1 Perpendicular Tangent Theorem Words; If a line is tangent to a circle then it is perpendicular to the radius drawn at the point of tangency.. B ℓ C Symbols; If ℓ is tangent to ⊙ C at B, then ℓ ⊥ .

Theorem 11.2 Perpendicular Tangent Converse Words; In a plane, if a line is perpendicular to a radius of a circle at its endpoint on a circle, then the line is tangent to the circle. B ℓ C Symbols; If ℓ ⊥ , then ℓ is tangent to ⊙ C at B.

Example 1 AC is tangent to B at point C. Find BC. SOLUTION is a radius of B, so you can apply Theorem 11.1 to conclude that and are perpendicular. BC AC So, BCA is a right angle, and ∆BCA is a right triangle. To find BC, use the Pythagorean Theorem.

Subtract 144 from each side. Find the positive square root. Example 1 (BA)2 = (BC)2 + (AC)2 Pythagorean Theorem 132 = (BC)2 + 122 Substitute 13 for BA and 12 for AC. 169 = (BC)2 + 144 Multiply. 25 = (BC)2 Subtract 144 from each side. 5 = BC Find the positive square root.

Example 2 You are standing at C, 8 feet from a silo. The distance to a point of tangency is 16 feet. What is the radius of the silo? SOLUTION Tangent is perpendicular to radius at B, so ∆ABC is a right triangle. So, you can use the Pythagorean Theorem. AB BC

The radius of the silo is 12 feet. Example 2 (AC)2 = (AB)2 + (BC)2 Pythagorean Theorem (r + 8)2 = r2 + 162 Substitute r + 8 for AC, r for AB, and 16 for BC. r2 + 16r + 64 = r2 + 256 (r + 8)(r + 8) = r2 + 16r + 64 16r + 64 = 256 Subtract r2 from each side. 16r = 192 Subtract 64 from each side. r =12 Divide each side by 16. ANSWER The radius of the silo is 12 feet.

How can you show that must be tangent to D? EF Example 3 Verify a Tangent to a Circle How can you show that must be tangent to D? EF SOLUTION Use the Converse of the Pythagorean Theorem to determine whether ∆DEF is a right triangle. Compare (DF)2 with (DE)2 + (EF)2. (DF)2 (DE)2 + (EF)2 = ? Substitute 15 for DF, 9 for DE, and 12 for EF. 152 92 + 122 = ? Multiply. 225 81 + 144 = ? 225 = 225 Simplify.

Example 3 Verify a Tangent to a Circle ∆DEF is a right triangle with right angle E. So, is perpendicular to . By Theorem 11.2, it follows that is tangent to D. EF DE

Example 4: ALGEBRA is tangent to at point R. Find y. Because the radius is perpendicular to the tangent at the point of tangency, . This makes a right angle and  a right triangle. Use the Pythagorean Theorem to find QR, which is one-half the length y.

Example 4: Pythagorean Theorem Simplify. Subtract 256 from each side. Take the square root of each side. Because y is the length of the diameter, ignore the negative result. Answer: Thus, y is twice .

Your Turn: is a tangent to at point D. Find a. Answer: 15

Example 5a: Determine whether is tangent to First determine whether ABC is a right triangle by using the converse of the Pythagorean Theorem.

Example 5a: Pythagorean Theorem Simplify. Because the converse of the Pythagorean Theorem did not prove true in this case, ABC is not a right triangle. Answer: So, is not tangent to .

Example 5b: Determine whether is tangent to First determine whether EWD is a right triangle by using the converse of the Pythagorean Theorem.

Example 5b: Pythagorean Theorem Simplify. Because the converse of the Pythagorean Theorem is true, EWD is a right triangle and EWD is a right angle. Answer: Thus, making a tangent to

Your Turn: a. Determine whether is tangent to Answer: yes

Your Turn: b. Determine whether is tangent to Answer: no

Tangent Segments A Tangent Segment touches a circle at one of the segment’s endpoints and lies in the line that is tangent to the circle at that point. C A B Tangent Segment

Theorem 11.3 Congruent Tangent Segments Words; If two segments from the same point outside a circle are tangent to the circle, then they are congruent. R S T Symbols;

Example 6 Find the value of x. is tangent to C at B. is tangent to C at D. AB AD SOLUTION AD = AB Two tangent segments from the same point are congruent. 2x + 3 = 11 Substitute 2x + 3 for AD and 11 for AB. 2x = 8 Subtract 3 from each side. x = 4 Divide each side by 2.

and are tangent to A. Find the value of x. Your Turn: and are tangent to A. Find the value of x. CB CD 1. 15 ANSWER 2. ANSWER 3

Example 7: ALGEBRA Find x. Assume that segments that appear tangent to circles are tangent. are drawn from the same exterior point and are tangent to so are drawn from the same exterior point and are tangent to

Example 7: Definition of congruent segments Substitution. Use the value of y to find x. Definition of congruent segments Substitution Simplify. Subtract 14 from each side. Answer: 1

Your Turn: ALGEBRA Find a. Assume that segments that appear tangent to circles are tangent. Answer: –6

Example 8: Triangle HJK is circumscribed about Find the perimeter of HJK if

Example 8: Use Theorem 10.10 to determine the equal measures. We are given that Definition of perimeter Substitution Answer: The perimeter of HJK is 158 units.

Your Turn: Triangle NOT is circumscribed about Find the perimeter of NOT if Answer: 172 units

Assignment Pg. 598 – 600; #1 – 39 odd