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The length of three concesutive sides of a quadrilateral circumscribing a circle are 4 cm, 5 cm, and 7 cm respectively. Determine the length of the fourth side.

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#### Solution

Let us first put the given data in the form of a diagram.

From the property of tangents we know that the length of two tangents drawn from the same external point will be equal. Therefore we have,

*AR = SA*

Let us represent *AR* and *SA* by ‘*a’*.

Similarly,

*QB = RB*

Let us represent SD and DP by *‘b’*

*PC = CQ*

Let us represent *PC* and *PQ* by ‘*c’*

*SD = DP*

Let us represent *QB* and *RB* by ‘*d*’

It is given that,

*AB* = 4

*AR + RB* =4

*a + b* = 4

*b* = 4 − *a* …… (1)

Similarly,

*BC* = 5

That is,

*b + c* = 5

Let us substitute for *b* from equation (1). We get,

4 − *a + c* = 5

*c − a* = 1

*c = a* + 1 …… (2)

*CD* = 7

*c + d* = 7

Let us substitute for *c* from equation (2). We get,

*a* + 1 + *d* = 7

*a + d *= 6

In the previous section we had represented AS and SR with ‘*a*’ and SD and DP with ‘*b*’. We shall now put AS in place of ‘*a*’ and SD in place of ‘*d*’. We get,

*AS + SD* = 6

*AD* = 6 cm

Therefore, the length of the fourth side of the quadrilateral is 6 cm.

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