How to Find the Tangent of a Circle | Geometry | Study.com - Free Printable

Let's solve the problem step by step.

---

Given:

- Triangle $ \triangle ABC $ with a circle inscribed (or tangent) at points $ B $ and $ C $.
- $ AB = AC $ → triangle is isosceles with $ AB = AC $
- $ AB = 12 $
- $ AC = x^2 - 4 $
- The segments from $ A $ to the points of tangency $ B $ and $ C $ are equal in length, and both are tangent to the circle.
- $ OB = OC = 9 $, where $ O $ is the center of the circle.
- $ OB $ and $ OC $ are radii, and they are perpendicular to the tangents at $ B $ and $ C $, respectively.

We are asked to:
1. Find the value of $ x $
2. Find the measure of $ AO $

---

Step 1: Use the given $ AB = AC $

We are told:
$$
AB = AC
$$
But:
- $ AB = 12 $
- $ AC = x^2 - 4 $

So:
$$
x^2 - 4 = 12
$$

Solve for $ x $:
$$
x^2 = 16 \\
x = \pm 4
$$

Since $ x $ represents a length or a variable in a geometric context (and $ x^2 - 4 $ must be positive), we take the positive root:
$$
\boxed{x = 4}
$$

✔ Answer to part 1: $ x = 4 $

---

Step 2: Find the measure of $ AO $

We need to find the length of segment $ AO $, from vertex $ A $ to the center $ O $ of the circle.

From the diagram:
- $ OB = 9 $, and $ OB \perp AB $ (since radius is perpendicular to tangent)
- So triangle $ ABO $ is a right triangle, right-angled at $ B $
- $ AB = 12 $
- $ OB = 9 $
- $ AO $ is the hypotenuse of right triangle $ \triangle ABO $

Use the Pythagorean Theorem:

$$
AO^2 = AB^2 + OB^2 \\
AO^2 = 12^2 + 9^2 = 144 + 81 = 225 \\
AO = \sqrt{225} = 15
$$

✔ Answer to part 2: $ AO = 15 $

---

✔ Final Answers:

1. $ x = 4 $
2. $ AO = 15 $

---

Let me know if you'd like a diagram explanation or further clarification!