To find the length of the chord of the larger circle that touches the smaller circle, we can use geometry to set up the problem.

Key Information:

• The two circles are concentric, meaning they share the same center.
• The radius of the larger circle is $R = 5 \, \text{cm}$.
• The radius of the smaller circle is $r = 3 \, \text{cm}$.
• The chord of the larger circle touches the smaller circle. This means the distance from the center to the chord is equal to the radius of the smaller circle, $r = 3 \, \text{cm}$.

Step-by-Step Solution:

1. Draw a diagram: Consider the center of the circles as $O$, the midpoint of the chord as $P$, and the points where the chord intersects the larger circle as $A$ and $B$.

2. Use a right triangle:

   • The line segment $OP$ (from the center $O$ to the midpoint $P$ of the chord) is perpendicular to the chord and has a length equal to the radius of the smaller circle, i.e., $OP = 3 \, \text{cm}$.
   • The radius of the larger circle is $OA = OB = 5 \, \text{cm}$.
   • Triangle $OAP$ is a right triangle where $OA = 5 \, \text{cm}$, $OP = 3 \, \text{cm}$, and we need to find $AP$, which is half of the length of the chord.

3. Apply the Pythagorean theorem: In the right triangle $OAP$: $OA^2 = OP^2 + AP^2$ Substituting the values: $5^2 = 3^2 + AP^2$ $25 = 9 + AP^2$ $AP^2 = 25 - 9 = 16$ $AP = \sqrt{16} = 4 \, \text{cm}$

4. Find the length of the chord: Since $AP = 4 \, \text{cm}$, the full length of the chord $AB$ is: $AB = 2 \times AP = 2 \times 4 = 8 \, \text{cm}$

Final Answer:

The length of the chord of the larger circle that touches the smaller circle is $8 \, \text{cm}$.

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Related Questions:

1. What is the length of a chord in a circle if the perpendicular distance from the center is given?
2. How do you find the distance between two concentric circles?
3. What is the formula for the length of a chord at a given distance from the center?
4. How do you derive the equation of a circle from its radius and center?
5. How would the problem change if the two circles were not concentric?

Tip:

In problems involving chords and circles, using the Pythagorean theorem in conjunction with radii and perpendiculars to the chord is a common and effective approach!