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Construct a direct common tangent to two circles of radii 4 cm and 2 cm whose centers are 8 cm apart. Measure and verify the length of the tangent.

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Last updated date: 29th Mar 2024
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MVSAT 2024
Answer
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Hint: For solving this problem, first line segment and then by using this line segment draw two circles of given radii. By using the difference between the radii construct another circle. Now, draw perpendicular to the initial line segment. Join the endpoints to the circle. Using this methodology, we can construct the tangent.

Complete Step-by-Step solution:
According to our problem, we are required to draw the direct common tangent to two circles whose radii are provided. First, draw a line segment AB of length 15cm. Using this line segment, draw circle C1 with radius 4cm and center as A. Again, draw another circle C2 with radius 2cm and center B.
Now, draw circle C3 with radius as the difference of the radii of A and B i.e., (R =4−2=2cm) and center A. B is the external point to circle C3 with center A. For finding the mid-point of AB, draw a perpendicular bisector of AB. Now, the intersection of AB and bisector is the mid-point M.
Taking M as center, construct a circle C4 having radius AM. Mark the point of intersection of C4 and C3 as E and F. Join AE and AF. Extend AE so that it meets C1 at P and AF meet C1 at R. now, draw BQ parallel to AP and BS parallel to AR.
Join PQ and RS to obtain the required direct tangents to initial given circles.
Foe verification: The length of the tangent will be:
$\begin{align}
  & PQ=\sqrt{{{d}^{2}}-{{\left( {{r}_{1}}-{{r}_{2}} \right)}^{2}}} \\
 & PQ=\sqrt{{{8}^{2}}-{{(4-2)}^{2}}} \\
 & PQ=\sqrt{64-4}=\sqrt{60} \\
 & \therefore PQ\approx 7.75 \\
\end{align}$
This length is similar to the length as obtained from the figure.
This could be roughly shown as:
seo images


Note: Students must follow the similar procedure to obtain the required tangent. They must not forget to verify the length of the tangent with the actual length because the question demands to verify the length also.
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