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Circle B is internally tangent to circle A at point P. Likewise, circle C is internally tangent to circle D at point P. If BD = AC, AB = 4, and SD = 2, find the length of the diameters for circles C and D. Be sure to justify your work. (Hint: BD = BC + CD and AC = AB + BC by the segment addition postulate.

Sagot :

AshrieIIeidn

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[tex]\rm{\pink{Justification:}}[/tex]

[tex]\sf\pink{ᯓ★}[/tex] The given information states that circle B is internally tangent to circle A at point P, and circle C is internally tangent to circle D at point P. This means that the points of tangency are the same point, P.

[tex]\sf\pink{ᯓ★}[/tex] The segment addition postulate states that if a point lies between two other points on a line segment, then the length of the entire segment is equal to the sum of the lengths of the two parts.

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[tex]\sf\pink{ᯓ★}[/tex] Using the segment addition postulate, we have:

  • BD = BC + CD
  • AC = AB + BC

[tex]\sf\pink{ᯓ★}[/tex] Substituting the given values:

  • BD = AC (given)
  • AB = 4 (given)
  • SD = 2 (given)

[tex]\sf\pink{ᯓ★}[/tex] Solving for BC:

  • BD = BC + CD
  • AC = AB + BC
  • 4 = 4 + BC
  • BC = 0

[tex]\sf\pink{ᯓ★}[/tex] Since BC = 0, the diameter of circle C is equal to the chord AC, which is 4.

[tex]\sf\pink{ᯓ★}[/tex] Again using the segment addition postulate:

  • BD = BC + CD
  • 2 = 0 + CD
  • CD = 2

[tex]\sf\pink{ᯓ★}[/tex] The diameter of circle D is twice the length of CD, which is the radius of circle D.

  • Diameter of circle D = 2 × CD = 2 × 2 = 4

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Therefore, the length of the diameter for circle C is [tex]\blue{\underline{\sf\pink{4}}}[/tex], and the length of the diameter for circle D is also [tex]\blue{\underline{\sf\pink{4}}}[/tex].

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