M and N are the centres of two circles with radii 12 cm and 8 cm respectively. QR is the common tangent to the circles. If the distance MN between the centres is 16 cm, find the length of the common tangent QR.

Question

Accepted Answer

Given two circles with centers M and N, radii 12 cm and 8 cm respectively, and the distance MN = 16 cm.

We are to find the length of the common tangent QR.

Since the circles do not intersect and the tangent is external, the length of the common tangent (l) can be found using the formula:

l = √(MN^2 - (r1 - r2)^2)

Where r1 and r2 are the radii of the two circles.

Step 1: Calculate (r1 - r2) = 12 - 8 = 4 cm
Step 2: Calculate (r1 - r2)^2 = 4^2 = 16
Step 3: Calculate MN^2 = 16^2 = 256  
Step 4: Calculate l = √(256 - 16) = √240
Step 5: Simplify √240 = √(16 × 15) = 4√15 cm

However, the problem states QR is the common tangent, and the options include 4√15 cm but the correct formula for the length of the direct common tangent between two circles with centers M and N and radii r1 and r2 is:

Length of direct common tangent = √(MN^2 - (r1 + r2)^2)

But since the circles are separate and the tangent is external, the length of the common tangent is:

Length = √(MN^2 - (r1 - r2)^2)

So the calculation above is correct.

Therefore, the length of QR = 4√15 cm.

Let's verify carefully.

Given:
r1 = 12 cm
r2 = 8 cm
MN = 16 cm

Calculate (r1 - r2)^2 = (12 - 8)^2 = 4^2 = 16
Calculate MN^2 = 16^2 = 256

Length of common tangent = √(MN^2 - (r1 - r2)^2) = √(256 - 16) = √240 = 4√15 cm

Hence, the correct answer is 4√15 cm.

Therefore, option 2 is correct.

Answer

Answer

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