Theorem 3: Theorem 4

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Chapter 14: Circle Segment Relationships Topic 3: Segments Theorems - Tangents

Theorem 3: If a line is tangent to a circle,

it is perpendicular to the radius drawn to 𝐼𝐹: 𝐴𝐵̅̅ ̅̅ is a tangent the point of tangency. 𝐷 is a point of tangency 𝑇𝐻𝐸𝑁: 𝑂𝐷̅̅ ̅̅ ⊥ 𝐴𝐵̅̅ ̅̅

Theorem 4: Tangent segments to a circle from the same external point are congruent.

IF: 𝐴𝐵̅̅ ̅̅ is tangent to circle 𝑂 at 𝐴 𝐶𝐵̅̅ ̅̅ is tangent to circle 𝑂 at 𝐶

Then: 𝐴𝐵̅̅ ̅̅ ≅ 𝐶𝐵̅̅ ̅̅ Practice: 1.) 𝐶𝐵̅̅ ̅̅ is a tangent. Find x. 2.) 𝐶𝐵̅̅ ̅̅ is a tangent. Find x. 3.) In the diagram, is 𝐶𝐵̅̅ ̅̅ a tangent? 4.) 𝐴𝐶̅̅ ̅̅ is a diameter, the radius = 9, 𝐶𝐵̅̅ ̅̅ = 24, 𝐴𝐵̅̅ ̅̅ = 30. Is 𝐶𝐵̅̅ ̅̅ a tangent?

5.) 𝐴𝐵̅̅ ̅̅ , 𝐶𝐵̅̅ ̅̅ are tangents. Find x. 6.) 𝐴𝐵̅̅ ̅̅ , 𝐶𝐵̅̅ ̅̅ are tangents. Find x. 7.) Find the perimeter of ∆𝐴𝐵𝐶 8.) Find the perimeter of ∆𝐴𝐵𝐶 Common Core Questions: 9.) In circle A, the radius is 9 mm and BC = 12 mm. (a) Find AC. (b) Find the area of ∆𝐴𝐶𝐷. (c) Find the perimeter of quadrilateral ABCD.

10.) In circle A, EF = 12 and AE = 13. AE:AC = 1:3. (a) Find the radius of the circle. (b) Find BC (round to the nearest whole number). (c) Find EC.

11.) 𝐵𝐶̅̅ ̅̅ is tangent to circle A at point B. DC = 9 and BC = 15. (a) Find the radius (r) of the circle. (b) Find AC.

12.) In the diagram, what do you think the length of z could be? How do you know? 13.) In the figure given, the three segments are tangent to the circle at points F, B, and G. Find DE. Explain how you arrived at your answer.

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Segments Theorems - Tangents HOMEWORK

Complete the following questions below. Show all work, including formulas.

1.) In the diagram below, circles X and Y have two tangents drawn to them from external point T. The points of tangency are C, A, S, and E. The ratio of TA to AC is 1:3. If TS = 24, find the length of SE. 2.) Mrs. Smith wants to purchase a cover for their new circular pool. She needs to know the radius of the pool, but she does not want to get wet to take the measurements. She is standing 4 feet from the pool and 12 feet from the point of tangency. Find the radius of the pool. 3.) In the diagram below, 𝐴𝐵̅̅ ̅̅ , 𝐵𝐶̅̅ ̅̅ , and 𝐴𝐶̅̅ ̅̅ are tangents to circle O at points F, E, and D, respectively, AF = 6, CD = 5, and BE = 4. What is the perimeter of ΔABC? (1) 15 (2) 25

(3) 30 (4) 60 4.) As shown in the diagram below, 𝐵𝑂̅̅ ̅̅ and tangents 𝐵𝐴̅̅ ̅̅ and 𝐵𝐶̅̅ ̅̅ are drawn from an external point B to circle O.

Radii 𝑂𝐴̅̅ ̅̅ and 𝑂𝐶̅̅ ̅̅ are drawn. If OA = 7 and DB = 18, determine and state the length of 𝐴𝐵̅̅ ̅̅ .

5.) The radius of circle A is 4. 𝐷𝐶̅̅ ̅̅ and 𝐶𝐸̅̅̅̅ are tangent to the circle with DC = 12. (a) Find the length of 𝐴𝐵𝐶̅̅ ̅̅ ̅̅ in simplest radical form. (b) Find the area of quadrilateral ADCE to the nearest hundredth.

6.) In the diagram below, 𝐴𝐶̅̅ ̅̅ and 𝐵𝐶̅̅ ̅̅ are tangent to circle O at A and B, respectively, from external point C. If 𝑚∠𝐴𝐶𝐵 = 38, what is 𝑚∠𝐴𝑂𝐵? (1) 71 (2) 104 (3) 142 (4) 161 7.) In the diagram below, 𝐴𝐶̅̅ ̅̅ and 𝐴𝐷̅̅ ̅̅ are tangent to circle B at points C and D, respectively, and 𝐵𝐶̅̅ ̅̅ , 𝐵𝐷̅̅ ̅̅ , and 𝐵𝐴̅̅ ̅̅ are drawn. (a) If AC = 12 and AB = 15, what is the length of 𝐵𝐷̅̅ ̅̅ ? (b) What is the perimeter of quadrilateral ACBD?

Review Questions: 8.) Triangle ABC had vertices A(0,0), B(6,8), and C(8,4). Which equation represents the perpendicular bisector of 𝐵𝐶̅̅ ̅̅ ?

(1) y = 2x – 6 (2) y = -2x + 4 (3) 𝑦 =1

2𝑥 +

5

2 (4) 𝑦 = −

1

2𝑥 +

19

2

9.) A student used a compass and a straightedge to construct 𝐶𝐸̅̅̅̅ in ∆𝐴𝐵𝐶 as shown below. Which statement must always be true for this construction? (1) ∠𝐶𝐸𝐴 ≅ ∠𝐶𝐸𝐵 (2) ∠𝐴𝐶𝐸 ≅ ∠𝐵𝐶𝐸 (3) 𝐴𝐸̅̅ ̅̅ ≅ 𝐵𝐸̅̅ ̅̅

(4) 𝐸𝐶̅̅̅̅ ≅ 𝐴𝐶̅̅ ̅̅ 10.) Quadrilateral ABCD is graphed on the set of axes below. Classify quadrilateral ABCD. Explain your reasoning.