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Chap. 4.: Geometric Constructions

) Construct two concentric circles with centre 'O' with radii 3 cm and 5 cm. Construct tangent to a smaller circle from any point A on the larger circle. Measure and write the length of tangent segment. Calculate the length of tangent segment using Pythagoras theorem.

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.

Construct two concentric circles with centre 'O' with radii 3 cm and 5 cm. Construct tangent to a smaller circle from any point A on the larger circle. Measure and write the length of tangent segment. Calculate the length of tangent segment using Pythagoras theorem.

Explore how to construct two concentric circles with radii 3 cm and 5 cm, and find the length of the tangent from a point on the larger circle to the smaller circle using the Pythagorean theorem. Suitable for Grades 6-8 students.

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