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The length of the diameter of ⊙M is 76 cm and the length of the diameter of ⊙J is 64 cm. If the length of JK is 12 cm, what is the length of LM

User EjLev
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Answer:

We can start by drawing a diagram of the situation. Let O be the center of both circles and let K be a point on the circumference of ⊙J such that JK = 12 cm. Let L be the point where JK intersects ⊙M, and let M be the point diametrically opposite L on ⊙M. Then, we have a right triangle JOK with legs JO = 32 cm and OK = 38 cm, and a right triangle LOM with legs LO = OM = r, where r is the radius of ⊙M. The hypotenuse of both triangles is the same and has length 64 cm.

We can use the Pythagorean theorem to find r. In the right triangle JOK, we have:

JO^2 + OK^2 = JK^2

32^2 + 38^2 = JK^2

JK = sqrt(32^2 + 38^2) ≈ 49.21 cm

In the right triangle LOM, we have:

LO^2 + OM^2 = LM^2

r^2 + r^2 = (2r)^2

2r^2 = LM^2

We know that LM = 2r + 12, since JK = 12 cm. Substituting this into the equation above, we get:

2r^2 = (2r + 12)^2

2r^2 = 4r^2 + 48r + 144

2r^2 - 4r^2 - 48r - 144 = 0

-r^2 - 24r - 72 = 0

r^2 + 24r + 72 = 0

We can solve for r using the quadratic formula:

r = (-24 ± sqrt(24^2 - 4*72)) / 2

r = (-24 ± sqrt(384)) / 2

r = -12 ± 4sqrt(6)

Since r is the radius of ⊙M, we want the positive value of r. Therefore:

r = -12 + 4sqrt(6) ≈ 4.03 cm

Finally, we can find LM:

LM = 2r + 12

LM = 2(4.03) + 12

LM ≈ 20.06 cm

Therefore, the length of LM is approximately 20.06 cm.

User Tetris
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