Question

In the above figure, Quadrilateral ABCD is circumscribing a circle with O and Ad perpendicular to AB. if radius of incircle is 10 cm, then value of x is ?​

Answer 1

$\huge✎\fbox\orange{QUE} {\colorbox{blue}{ST}} \fbox\green{ION}☟$

In the above figure, Quadrilateral ABCD is circumscribing a circle with O and Ad perpendicular to AB. if radius of incircle is 10 cm, then,

$\orange{\large{\mathcal\colorbox{indigo}{Find the value of x}}}$

$\huge✍︎\fbox\orange{ÂŇ}\fbox{SW} \fbox\green{ÊŘ}:$

Brother/Sister your correct answer will be:-

$\orange{\huge{\mathcal\colorbox{navy}{21 cm}}}$

Step-by-step explanation:-

$\huge \underline \purple{Construction:-}$

• Join RO
• Join OP
• Join OQ

$\huge\underline \purple{Solution :-}$

Here, x = AB = AP + PB. We need value of AP and PB.

Beacuse we don't have direct method for finding value of x.

$\purple{We \: know,}$

• Tangent at any point of circle is perpendicular to the radius through point of contact.

$\purple{So,}$

$\red{ \: \: \: • \: \: ∠OSA = ∠OPA = 90°}$

$\red{And, ∠A = 90°} \green {[Given]}$

$\pink{\leadsto ∠OSA = ∠OPA = ∠A = 90°}$

If three angles of quadrilateral ASOP are 90°. Then,

$\pink{∠SOP \: will \: also \: 90°}$

Thus, Quadrilateral ASOP is a square.

All sides of Square are equal.

$\purple{So,}$

$\leadsto AP = 10 cm$

ROQC is a quadrilateral and C is an exterior point of circle.

$\purple{We \: know,}$

Lengths of tangents drawn from an external point to a circle are equal.

$\purple{So,}$

$\pink {\leadsto RC = QC = 27 \: cm}$

$\purple{Similarly,}$

In quadrilateral POQB, B is an external point.

$\purple{So,}$

$\pink{ \: \: \: \: \: • \: \: PB = QB}$

$\pink{ \leadsto QB = BC - CQ}$

$\pink{ \leadsto QB = 38 - 27}$

$\pink{\leadsto QB = 11}$

$\pink{QB = PB}$

$\purple{So,}$

$\pink{PB = 11 cm}$

$\purple{Now,}$

$\pink{\leadsto x = AP + PB}$

$\pink{\leadsto x = 10 + 11}$

$\pink{\leadsto x = 21}$

$\purple{Therefore,}$

$\red{ \underline\green {\colorbox{orange}{Value \: of \: x \: is \: 21 \: cm}}}$

$\red{꧁Hopes\:so\:that\:this\:will\:help\:you,꧂}$

$\huge\colorbox{orange}❤\colorbox{indigo}{\red{With\:Love}}\colorbox{orange}❤$

✌︎Thanking you, your one of the Brother of your 139 Crore Indian Brothers and Sisters.✌︎

$\large\red{\underbrace{\overbrace\mathbb{\colorbox{aqua}{\colorbox{blue}{\colorbox{purple}{\colorbox{pink}{\colorbox{green}{\colorbox{gold}{\colorbox{silver}{\textit{\purple{\underline{\red{《❀PŘÎ᭄ŇCĒ✿》࿐}}}}}}}}}}}}}}$