(4)

Draw CM perpendicular to AB.

Now, AM = MC and MB = MC (tangents drawn from external point are equal).

⇒ AM = MC

⇒ $\angle MAC$ = $\angle MCA$ = 45°

(Since Δ AMC is right triangle)

∴  Also, MB = MC ⇒ $\angle MBC$ = $\angle MCB$ = 45° (Since Δ MBC is right angle triangle)

∴ $\angle ACB$ = $\angle MCA$ + $\angle MCB$ = 45° + 45° = 90° ⇒ $\angle ACB$ = 90°

(2)

Area of the circle > Area of the square

(2)

Assertion: Two parallel tangents always lie at the end points of the diameter of the circle.

Reason: Diameter is the longest chord of a circle which passes through centre joining the two points on the circumference of a circle.

Both, Assertion (A) and Reason (R) are true but Reason (R) is not correct explanation for Assertion (A).

(4)

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ABCD is a square of side 6 cm. PQ is a diameter of the given circle such that $PQ=AB=6cm$

$\therefore$ $\text{Radius }\left(r\right)=\frac{6}{2}=3\text{ cm}$

$\text{Area of the circle}=\pi r^2=\pi {\left(3\right)}^2=9\pi {\text{ cm}}^2.$

(4)

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Angle between two tangents = $60°$     (given)

$\therefore$  Tangents are equally inclined to each other

$\therefore$  $\angle OPA=\angle OPB=30°$

and $\angle OAP=90°$

(Tangent is perpendicular to the radius)

In $∆PAO,$ $\tan 30°=\frac{OA}{AP}\Rightarrow \frac{1}{\sqrt{3}}=\frac{3}{AP}\Rightarrow AP=3\sqrt{3}$

Hence, the length of each tangent is $3\sqrt{3}$ cm

(2)

PA = PB (Tangents from an external point P)

$∆\mathrm{APB}$ is a Right angle Triangle

$A{B}^2=A{P}^2+P{B}^2$$\text{[AP = PB]}$

$A{B}^2=2A{P}^2$

$A{B}^2=2\times 5^2$

$\Rightarrow AB=5\sqrt{2}\text{ cm}$

(2)     $5\sqrt{3}cm$

(4)

$\angle APD=\angle CPB$ [vertically opposite angle]

$\angle ADP=\angle CBP$ [Angle subtends on the same segment]

$∆ADP~∆CBP$ (by AA similarity)

(4)

Given, $\angle CAT=40°$

$\angle BAT=90°$

$\Rightarrow \angle BAC+\angle CAT=90°$

$\Rightarrow \angle BAC=50°$

$\Rightarrow \angle ACB=90° \text{[Angle in semi-circle]}$

$\text{In }∆ABC, \angle A+\angle B+\angle C=180°$

$\Rightarrow 50°+\angle B+90°=180°$

$\Rightarrow \angle B=180°-140°=40°$

(4)     Assertion (A) is false, but Reason (R) is true.