To solve this problem, we need to follow these steps:

Step-by-Step Solution:

1. Draw Two Concentric Circles:

   • First, draw a point $O$, which will serve as the center for both circles.
   • Draw a circle with radius $3 \, \text{cm}$ centered at $O$.
   • Then, draw another circle centered at $O$ with a radius of $5 \, \text{cm}$.
   • Now, you have two concentric circles: the smaller circle with radius $3 \, \text{cm}$ and the larger circle with radius $5 \, \text{cm}$.

2. Choose a Point on the Larger Circle:

   • Choose a point $A$ on the circumference of the larger circle (radius $5 \, \text{cm}$).

3. Construct the Tangent from Point $A$ to the Smaller Circle:

   • Draw a line from $A$ to $O$, which will serve as the radius to point $A$.
   • Now, we need to draw a tangent from $A$ to the smaller circle, touching it at one point only. Let the point of tangency on the smaller circle be $T$.
   • The line segment $AT$ is the tangent to the smaller circle from $A$.

4. Calculate the Length of the Tangent Segment $AT$:

   • Since $AT$ is tangent to the smaller circle from point $A$, we can use the Pythagorean theorem to calculate its length.
   • In the right triangle $\triangle AOT$:
     • $OA = 5 \, \text{cm}$ (radius of the larger circle),
     • $OT = 3 \, \text{cm}$ (radius of the smaller circle),
     • $AT$ is the tangent we want to find.

   Using the Pythagorean theorem: $AT = \sqrt{OA^2 - OT^2}$ Substitute the values: $AT = \sqrt{5^2 - 3^2}$ $AT = \sqrt{25 - 9}$ $AT = \sqrt{16}$ $AT = 4 \, \text{cm}$

Answer:

• The length of the tangent segment $AT$ is $4 \, \text{cm}$.

Would you like further details on the construction steps or on applying the Pythagorean theorem here?

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Related Questions:

1. How would the length of the tangent change if the radii of the circles were doubled?
2. Can you construct a tangent to the smaller circle from any point not on the larger circle?
3. What would the tangent length be if the radius of the smaller circle was $4 \, \text{cm}$ and the larger circle was $7 \, \text{cm}$?
4. How can we verify that $AT$ is perpendicular to the radius at point $T$?
5. What are some real-world applications of concentric circles and tangent lines?

Tip: When working with concentric circles, the radius difference can often reveal relationships about tangents and distances.