Consider the figure, which shows a circle with center at C and a radius of r, and right triangle ADC.
Find r if AB = 1 and AD = 5.
r =

To find the radius \( r \) of the circle, we need to use the given information and apply some geometric principles. 

### Step 1: Analyzing the Triangle and the Circle
Given:
- \( AB = 1 \)
- \( AD = 5 \)

Since \( \triangle ADC \) is a right triangle and \( C \) is the center of the circle, \( CD \) is perpendicular to \( AD \) and \( C \) lies on the hypotenuse \( AD \) (since the hypotenuse is the diameter of the circle).

### Step 2: Use the Power of a Point Theorem
The Power of a Point Theorem (or specifically, the secant-tangent theorem in this case) states:
\[
AB^2 = AD \cdot (AD - 2r)
\]

Plugging in the values:
\[
1^2 = 5 \cdot (5 - 2r)
\]
\[
1 = 25 - 10r
\]

### Step 3: Solve for \( r \)
Rearrange the equation to solve for \( r \):
\[
10r = 25 - 1
\]
\[
10r = 24
\]
\[
r = \frac{24}{10} = 2.4
\]

Thus, the radius \( r \) of the circle is \( \boxed{2.4} \).

Would you like more details or have any questions? Here are five related questions you might find interesting:

1. How would the radius change if \( AB = 2 \) and \( AD = 6 \)?
2. Can you find the length of \( CD \) in the given scenario?
3. What happens to the radius if the triangle is not a right triangle but an obtuse one?
4. How would the problem change if the center \( C \) was not on the hypotenuse?
5. Can you derive a general formula for \( r \) in terms of \( AB \) and \( AD \)?

**Tip:** Understanding the relationship between the triangle's properties and the circle's geometry is crucial when working with inscribed or circumscribed figures.

Learn how to find the radius of a circle given a right triangle and segment lengths AB = 1 and AD = 5. This problem involves applying the Power of a Point Theorem to relate the triangle's properties to the circle's geometry. Suitable for students in Grades 10-12.

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