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# Construct a tangent to a circle of radius 4 cm from a point on the concentric circle of radius 6 cm and measure its length. Also verify the measurement by actual calculation.

Steps to construct the required figure:
Step 1: Draw concentric circles C and C’ of radius 4cm and 6cm respectively and mark the center as O.

Step 2: Mark two points as A and P such that O,A and P lie on the same line.

Step 3:Draw perpendicular bisector of OP which intersects OP at O’.

Step 4: By taking O’ as the center, draw a circle of radius OO’ which intersects the circle C at point T and Q.

Step 5: Now join PT and PQ.

Step 6: PT and PQ are our required tangents whose length is 4.5cm approx. JUSTIFICATION:

Join OT and OQ,

We have OT ⊥ PT [Radius ⊥ to tangent ]

In the right angled triangle OTP,

$${OP}^2={OT}^2+{PT}^2$$

$${(6)}^2={(4)}^2+{(PT)}^2$$

$${PT}^2=36-16=20$$

$$PT=\sqrt{20}=2\sqrt{5} cm$$

Similarly $$PQ=2\sqrt{5} cm$$

A pair of tangents can be drawn from an external point outside the circle and these tangents have equal length.
$$\therefore$$ PT=PQ