Construction of Tangents to a Circle

Draw a circle of radius $4$ cm which touches a given line $AB$ and a given circle of radius $3$ cm with centre $C$.

Draw a circle of radius $3$ cm which touches a given line $AB$ and a given circle of radius $2$ cm with centre $C$.

Construct an angle of $65°$ and draw a circle of radius $2.5$ cm  touching the arms of the angle.

Which of the following diagram is correct for construction of an angle of $70°$ and  a circle of radius $2.5$ cm  touching the arms of the angle.

To construct a circle Touching a Line $AB$ at Given Point$T$  and Passing through a Given Point $P$ the following steps has to be followed

Step $1$: Draw a straight line $AB$

Step $2$: Mark point $T$ any where on the line $AB$ and choose random point $P$

Step $3$: Draw a perpendicular line $TQ$  to the line $AB$ at the point $P$

Step $4$: Join $TP$

Step $5$: Draw a bisection line to the line $TP$ and extend till it intersect $TQ$ at $O$

Step $6$: Use compass to draw a circle at centre $O$ and passing through $P$ and touches the line $AB$

Construction of a Circle Touching a Line $AB$ at Given Point$T$  and Passing through a Given Point $P$

Step $1$: Draw a straight line $AB$

Step $2$: Mark point $T$ any where on the line $AB$ and choose random point $P$

Step $3$: Draw a perpendicular line $TQ$  to the line $AB$ at the point $P$

Step $4$: Join $TP$

Step $5$: Draw a bisection line to the line $TP$ and extend till it intersect $TQ$ at $O$

Step $6$: Use compass to draw a circle at centre $O$ and passing through $P$ and touches the line $AB$

Which step is incorrect?

In a step of construction of inscribed triangle, It is necessary to construct all three perpendicular bisectors

In a step of construction of inscribed triangle, It is not necessary to construct all three perpendicular bisectors

The dimension of the rectangle is $4\times 16$.

What would be the length of a square, if you want to construct a square whose its area equal to the rectangle?

The dimension of the rectangle is $2\times 32$.

What would be the length of a square, if you want to construct a square whose its area equal to the rectangle?

Draw a transverse common tangent to two circles of radii $6cm$ and $2cm$ whose centres are $8cm$ apart.

Construct a transverse common tangent to two circles of radii $4cm$ and $3cm$ whose centres are $10cm$ apart.

Construct two circles of radii $4 \mathrm{cm}$ and $2.5 \mathrm{cm}$ and the distance between their centres is $9 \mathrm{cm}$. Construct a direct common tangent of the circles.

Draw a line segment $\mathrm{MN}$ of length $8.5 \mathrm{cm}$. Draw two circles of radii $4 \mathrm{cm}$ and $2.5 \mathrm{cm}$ with $\mathrm{M}$ and $\mathrm{N}$ as centre respectively. Construct a pair of tangents to each circle from the centre of the other circle.

Triangle $ABC$ is right-angled at $A$, and $AD$ is the altitude from $A$ to the hypotenuse $BC$.
Find $x$.

Draw a tangent to a circle at a point on the circle. 

Construct two circles of radii $5 \mathrm{cm}$ and $3 \mathrm{cm}$ and the distance between their centres is $8.5 \mathrm{cm}$. Construct a direct common tangent of the circles.

Draw a line segment $\mathrm{PQ}$ of length $7 \mathrm{cm}$. Draw two circles of radii $2 \mathrm{cm}$ and $3 \mathrm{cm}$ with $\mathrm{P}$ and $\mathrm{Q}$ as centre respectively. Construct a pair of tangents to each circle from the centre of the other circle.

Draw a circle of diameter $7 \mathrm{cm}$. Take two points $A$ and $B$ on either side of the extended diameter, each at a distance of $5 \mathrm{cm}$ respectively from the centre. Draw tangents to the circle from the points $A$ and $B$. Write steps of construction.

Construct a tangent to a circle of radius $6 \mathrm{cm}$ from a point $P$ on the concentric circle of radius $10 \mathrm{cm}$ and measure its length. Also verify the measurement by actual calculation.