To solve this problem, we need to find the length of the chord in the larger circle that touches the smaller circle. Here’s how we can approach the problem:

Step 1: Understand the Geometry

• The two circles have radii of 17 cm (larger circle) and 8 cm (smaller circle).
• The centers of both circles lie along the same line, and the circles are tangent to each other externally, meaning they touch at one point.
• The chord we are looking for lies in the larger circle and touches the smaller circle.

Step 2: Define Key Elements

• Let the radius of the larger circle be $R = 17$ cm.
• Let the radius of the smaller circle be $r = 8$ cm.
• The distance between the centers of the two circles is $R + r = 17 + 8 = 25$ cm.
• The chord in the larger circle will be perpendicular to the line connecting the centers of the two circles, and it will touch the smaller circle at exactly one point.

Step 3: Use the Right Triangle Geometry

We can use the right triangle formed by:

1. The radius of the larger circle ($R$),
2. The distance from the center of the larger circle to the point of tangency of the smaller circle ($R - r$),
3. Half of the length of the chord, which we need to find.

By applying the Pythagorean theorem to this right triangle, where:

• The hypotenuse is the radius of the larger circle ($R = 17$),
• One leg is the distance from the center of the larger circle to the tangency point on the smaller circle ($R - r = 17 - 8 = 9$).

The relationship between these elements is:

$\left( \frac{\text{chord length}}{2} \right)^2 + (R - r)^2 = R^2$

Substituting the known values:

$\left( \frac{\text{chord length}}{2} \right)^2 + 9^2 = 17^2$

$\left( \frac{\text{chord length}}{2} \right)^2 + 81 = 289$

$\left( \frac{\text{chord length}}{2} \right)^2 = 289 - 81 = 208$

$\frac{\text{chord length}}{2} = \sqrt{208}$

$\frac{\text{chord length}}{2} \approx 14.42 \, \text{cm}$

Therefore, the full length of the chord is approximately:

$\text{chord length} \approx 2 \times 14.42 = 28.84 \, \text{cm}$

Final Answer:

The length of the chord in the larger circle that touches the smaller circle is approximately 28.84 cm.

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Would you like more details or further clarification on any part of this solution?

Here are 5 related questions you might find interesting:

1. How would the chord length change if the radii of the circles were different?
2. How can the same method be used for concentric circles?
3. What is the area of the segment of the larger circle formed by the chord?
4. How do you find the length of a chord in a circle given the distance from the center?
5. What happens if the circles are internally tangent instead of externally tangent?

Tip: The geometry of tangent circles often involves using the Pythagorean theorem to relate distances between centers and chord lengths.