Question

In the adjoining figure a circle touches all the 4 sides of a quadrilateral ABCD with AB - 6cm. BC- 7 cm. and CD-4 cm. Find AD. I will mark brain list if you give fast amswer​

Answer 1

Answer:

Kindly view the attachment, CONCEPT: Adjacent Tangents are equal.

Step-by-step explanation:

Answer 2

Answer:

The length of AD is 3 cm.

Step-by-step-explanation:

In fig., in $\sf\:\square\:ABCD$,

$\left\begin{tabular}{c}\sf\:AB\:=\:6\:cm\\\\\sf\:BC\:=\:7\:cm\\\\\sf\:CD\:=\:4\:cm\:\:\:\end{tabular}\right\}\:\sf\:[\:Given\:]$

Now,

$\sf\:AP\:=\:AS$ ... [ Tangent segments drawn from an external point to a circle are congruent. ]

Let AP = AS = x cm ... ( 1 )

Similarly, we can say that

$\left\begin{array}{c}\sf\:BP\:=\:BQ\:=\:(\:2\:)\\\\\sf\:CQ\:=\:CR\:=\:(\:3\:)\\\\\sf\:RD\:=\:DS\:=\:(\:4\:)\end{array}\right\}\begin{minipage}{5cm}\sf\:[\:Tangent\:segments\:drawn\:from\:an\:external\:point\:to\:a\:circle\:are\:congruent\:]\end{minipage}$

But, AP = x ... [ From ( 1 ) ]

$\therefore\sf\:AB\:=\:AP\:+\:BP\:\:-\:-\:[\:A\:-\:P\:-\:B\:]\\\\\implies\sf\:6\:=\:x\:+\:BP\:\:-\:-\:[\:From\:given\:and\:(\:1\:)\:]\\\\\implies\boxed{\sf\:BP\:=\:(\:6\:-\:x\:)}$

Now,

$\sf\:BP\:=\:BQ\:=\:(\:6\:-\:x\:)\:\:-\:-\:[\:From\:(\:2\:)\:]\:\:\:-\:-\:(\:5\:)$

Now,

$\sf\:BC\:=\:BQ\:+\:CQ\:\:-\:-\:-\:[\:B\:-\:Q\:-\:C\:]\\\\\implies\sf\:7\:=\:(\:6\:-\:x\:)\:+\:CQ\:\:[\:From\:given\:and\:(\:5\:)\:]\\\\\implies\sf\:CQ\:=\:7\:-\:(\:6\:-\:x\:)\\\\\implies\sf\:CQ\:=\:7\:-\:6\:+\:x\\\\\implies\boxed{\sf\:CQ\:=\:(\:1\:+\:x\:)}$

Now,

$\sf\:CQ\:=\:CR\:=\:(\:1\:+\:x\:)\:\:-\:-\:(\:From\:(\:3\:)\:]\:\:-\:-\:(\:6\:)$

Now,

$\sf\:CD\:=\:CR\:+\:RD\:\:-\:-\:[\:C\:-\:R\:-\:D\:]\\\\\implies\sf\:4\:=\:(\:1\:+\:x\:)\:+\:RD\:\:-\:-\:[\:From\:given\:and\:(\:6\:)\:]\\\\\implies\sf\:RD\:=\:4\:-\:(\:1\:+\:x\:)\\\\\implies\sf\:RD\:=\:4\:-\:1\:-x\\\\\implies\boxed{\sf\:RD\:=\:(\:3\:-\:x\:)}$

Now,

$\sf\:RD\:=\:DS\:=\:(\:3\:-\:x\:)\:\:-\:-\:[\:From\:(\:4\:)\:]\:-\:-\:(\:7\:)$

Now,

$\sf\:AD\:=\:AS\:+\:DS\:\:\:-\:-\:[\:A\:-\:S\:-\:D\:]\\\\\implies\sf\:AD\:=\:x\:+\:(\:3\:-\:x\:)\:\:-\:-\:[\:From\:(\:1\:)\:and\:(\:7\:)\:]\\\\\implies\sf\:AD\:=\:\cancel{x}\:+\:3\:-\:\cancel{x}\\\\\implies\boxed{\red{\sf\:AD\:=\:3\:cm}}$

Additional Information:

1. Tangent to a circle:

When a line touches a circle in one and only point, then the line is tangent to the circle.

Tangent is always perpendicular to the radius of the circle.

2. Tangent segment Theorem:

When two tangents are drawn to a circle from a point which is in the exterior of the circle, then the segments formed by those tangents are of equal length.