Two concentric circles are of radii 5 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle.

Step 1: Visualize the setup

We have two concentric circles, meaning they share the same center point, O. Let the radius of the larger circle be $R = 5$ cm and the radius of the smaller circle be $r = 3$ cm. We need to find the length of a chord of the larger circle that is tangent to the smaller circle.

Step 2: Form a right-angled triangle

Let AB be the chord of the larger circle. Since this chord touches the smaller circle, it is tangent to the smaller circle at a point, say P. The radius of the smaller circle, OP, will be perpendicular to the chord AB at point P. Thus, $\triangle OPA$ is a right-angled triangle with the right angle at P.

Step 3: Apply the Pythagorean theorem

In the right-angled triangle $\triangle OPA$, OA is the hypotenuse, which is the radius of the larger circle ($R = 5$ cm). OP is the radius of the smaller circle ($r = 3$ cm). AP is half the length of the chord AB. We can use the Pythagorean theorem to find the length of AP.

Step 4: Calculate AP

Substitute the values $OA = 5$ cm and $OP = 3$ cm into the Pythagorean theorem. Calculate $5^2 = 25$ and $3^2 = 9$. Subtract 9 from 25 to find $AP^2$. Taking the square root of 16 gives us the length of AP, which is 4 cm.

Step 5: Calculate the length of the chord AB

Since the radius OP is perpendicular to the chord AB, it bisects the chord. Therefore, the length of the chord AB is twice the length of AP. Multiply AP by 2 to get the final length of the chord.

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