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Circle O and circle P, with radii 3 and 5, respectively, are both tangent to line l at H. Enter all possible lengths of OP, separated by commas.

 Feb 19, 2023
 #1
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Let's draw a diagram to visualize the problem:

 

We can see that the line segment connecting the centers of the two circles, OP, is perpendicular to the line l at H, the point of tangency. We can use this fact to find the length of OP using the Pythagorean theorem.

Let x be the length of OP. Then we have:

OH = 3 (the radius of circle O) PH = 5 (the radius of circle P)

By the Pythagorean theorem, we have:

x^2 = OH^2 + PH^2 = 3^2 + 5^2 = 9 + 25 = 34

 

Therefore, only possible lengths of OP is sqrt(34).

 Feb 19, 2023
 #2
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incorrect

Guest Feb 20, 2023

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