Maharashtra Board Solutions for Chapter: Circle, Exercise 6: Problem set 3

In the adjoining figure circles with centres $\mathrm{X}$ and $\mathrm{Y}$ touch each other at point $\mathrm{Z}$. A secant passing through $\mathrm{Z}$ intersects the circles at points $\mathrm{A}$ and $\mathrm{B}$ respectively. Prove that, radius $\mathrm{XA}\parallel$ radius $\mathrm{YB}$. Fill in the blanks and complete the proof.

Construction: Draw segments $\mathrm{XZ}$ and ........

Proof: By theorem of touching circles, points $\mathrm{X}, \mathrm{Z}, \mathrm{Y}$ are ......

$\therefore \angle \mathrm{XZA}\cong$......... opposite angles

let $\angle \mathrm{XZA}=\angle \mathrm{BZY}=\mathrm{a}$     ..........$\left(\mathrm{I}\right)$

Now $\mathrm{seg} \mathrm{XA}\cong seg\mathrm{XZ}$   .........(..........)

$\therefore \angle \mathrm{XAZ}=$..........$\mathrm{a}$        ............(isosceles triangle theorem) $\left(\mathrm{II}\right)$

Similarly, $seg\mathrm{YB}\cong$...........       ...........(..............)

$\therefore \angle \mathrm{BZY}=$......... $=\mathrm{a}$   ...........(.........) $\left(\mathrm{III}\right)$

$\therefore$ From $\left(\mathrm{I}\right), \left(\mathrm{II}\right), \left(\mathrm{III}\right),$

$\angle \mathrm{XAZ}=$.........

$\therefore$ radius $\mathrm{XA}\parallel$ radius $\mathrm{YB}$ ..........(........)

In figure line $\mathrm{PR}$ touches the circle at point $\mathrm{Q}$. Answer the question with the help of the figure. 

$\angle TAS=65°$, find the measure of $\angle TQS$ and an arc $TS$?

In figure line $\mathrm{PR}$ touches the circle at point $\mathrm{Q}$. Answer the question with the help of the figure. 

$\angle TAS=65°$, find the measure of $\angle TQS$ and an arc $TS$?

In the figure, if $\mathrm{AB}=3.6 \mathrm{units}, \mathrm{AC}=9.0 \mathrm{units}$, $\mathrm{AD}=5.4 \mathrm{units}$, find $\mathrm{AE}$.

In figure, chord $\mathrm{EF}\parallel$ chord $\mathrm{GH}$. Prove that, chord $\mathrm{EG}\cong$ chord $\mathrm{FH}$. Fill in the blanks and write the proof.

Proof: Draw $\mathrm{seg} \mathrm{GF}$.

$\angle \mathrm{EFG}=\angle \mathrm{FGH}$ _____ $\left(\mathrm{I}\right)$

$\angle \mathrm{EFG}=$_____ inscribed angle theorem$\left(\mathrm{II}\right)$

$\angle \mathrm{FGH}=$_____ inscribed angle theorem$\left(\mathrm{III}\right)$

$\therefore \mathrm{m}(arc\mathrm{EG})=$_____ from $\left(\mathrm{I}\right), \left(\mathrm{II}\right), \left(\mathrm{III}\right).$

chord $\mathrm{EG}\cong$ chord $\mathrm{FH}$_____.

In figure, $\mathrm{P}$ is the point of contact of the tangent $\mathrm{OP}$ with the circle.

If $\mathrm{OP}=7.2,\mathrm{OR}=16.2$ find $\mathrm{QR}$.

In figure, $\mathrm{seg} \mathrm{AB}$ is a diameter of a circle with centre $\mathrm{O}$. The bisector of $\angle \mathrm{ACB}$ intersects the circle at point $\mathrm{D}$ Prove that, $seg\mathrm{AD}\cong seg\mathrm{BD}$. Complete the following proof by filling in the blanks.

Proof: Draw seg $\mathrm{OD}$.

$\angle \mathrm{ACB}=$_____ angle inscribed in a semicircle.

$\angle \mathrm{DCB}=$_____ $\mathrm{CD}$ is the bisector of $\angle \mathrm{C}$

$\mathrm{m}(arcDB)=$_____ inscribed angle theorem 

$\angle \mathrm{DOB}=$_____ definition of measure of an arc $\left(\mathrm{I}\right)$

$seg\mathrm{OA}\cong seg\mathrm{OB}$_____ $\left(\mathrm{II}\right)$

$\therefore$ line $\mathrm{OD}$ is _____ of $segAB$ _____ from $\left(\mathrm{I}\right)$ and $\left(\mathrm{II}\right)$

$\therefore seg\mathrm{AD}\cong seg\mathrm{BD}$

In the figure, $\mathrm{seg} \mathrm{MN}$ is a chord of a circle with centre $\mathrm{O}.$ $\mathrm{MN}=25$ units. $\mathrm{L}$ is a point on chord $\mathrm{MN}$ such that $\mathrm{ML}=9$ units and $\mathrm{d}(\mathrm{O},\mathrm{L})=5$ units. Find the radius of the circle in units.