Length of a common tangent of two circles: sqrt(d^2 - (r1 -/+ r2)^2)

This type of question, asked 3 times: CDS 2 2017, CDS 1 2018, CDS 1 2025. All 3 questions below, oldest first.

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Question 1
CDS 2 2017
66 $The distance between the centres of two circles having radii 9 cm and 4 cm is 13 cm. What is the length of the direct common tangent of these circles ?$
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Question 2
CDS 1 2018
73 $The radii of two circles are 4.5 cm and 3.5 cm respectively. The distance between the centres of the circles is 10 cm. What is the length of the transverse common tangent ?$
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Question 3 (case study, 2 parts)
CDS 1 2025
Consider the following for the next two (02) items that follow : A circle M of radius 8 cm touches externally with another circle N of radius 16 cm. Let P, Q be the points where the common tangent touches the circles M and N respectively.
55 $What is the length of the common tangent PQ ?$
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56 $If U, V are the centres of the circles M and N respectively, then what is the area of the quadrilateral formed by the points P, Q, V and U ?$
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Question 1