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.
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Step-by-Step Solution
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, △OPA is a right-angled triangle with the right angle at P.
Step 3: Apply the Pythagorean theorem
In the right-angled 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 52=25 and 32=9. Subtract 9 from 25 to find AP2. 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.