Multiple choice

Two circles, each with a radius of 7 cm, have a direct common tangent AB and an indirect common tangent CD. If the length of CD is 48 cm, find the length of AB.

  1. 25 cm

  2. 50 cm

  3. 55 cm

  4. 60 cm

  5. None of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

For two circles of equal radius r, the length of the direct common tangent AB is equal to the distance between centers d. The length of the indirect common tangent CD is sqrt(d^2 - (2r)^2). Given CD = 48 and r = 7, 48^2 = d^2 - 14^2. 2304 = d^2 - 196. d^2 = 2500. d = 50.

AI explanation

Since both circles have equal radii of 7 cm, the formula for the length of a direct common tangent is sqrt(d^2 - (R-r)^2) and for a transverse tangent is sqrt(d^2 - (R+r)^2). The term (R-r) equals zero, so the direct common tangent AB becomes equal to the distance between the centers, d. The transverse common tangent CD is given by the formula sqrt(d^2 - (R+r)^2), so substituting CD = 48 cm, R = 7 cm, and r = 7 cm gives 48 = sqrt(d^2 - 14^2), which simplifies to d^2 = 2304 + 196 = 2500, making d = 50 cm. Therefore, the length of AB is 50 cm.

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