Question

Two circles have radii of 12 cm and 4 cm. If the length of a direct common tangent is 15 cm, what is the distance between their centers?

• A. 15 cm
• B. 17 cm
• C. 18 cm
• D. 20 cm

Answer 1

Answer:

B. 17 cm

The formal formula for the length of a direct common tangent ($d$) between two circles is:

$d = \sqrt{D^2 - (R - r)^2}$

Where:

• $d$ = Length of the direct common tangent = 15 cm

• $D$ = Distance between the centers = ?

• $R$ = Radius of the larger circle = 12 cm

• $r$ = Radius of the smaller circle = 4 cm

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• Step 1: Substitute the given values into the formula
  $15 = \sqrt{D^2 - (12 - 4)^2}$

• Step 2: Simplify the radius difference
  $15 = \sqrt{D^2 - (8)^2}$
  $15 = \sqrt{D^2 - 64}$

• Step 3: Square both sides to remove the square root
  $15^2 = D^2 - 64$
  $225 = D^2 - 64$

• Step 4: Isolate $D^2$
  $D^2 = 225 + 64$
  $D^2 = 289$

• Step 5: Take the square root to solve for $D$
  $D = \sqrt{289}$
  $D = \mathbf{17\text{ cm}}$

The distance between their centers is 17 cm.