Alternate Segment and its Angles

A circle passes through the vertices of a $∆ABC$. The line $BC$ and the tangent to the circle at $A$ meet in $D$. If $BC=3 \mathrm{cm}, \mathrm{BD} = 4 \mathrm{cm},$ find $AD$.

In the given figure, $ABCD$ is a kite $\left(BC>AB\right)$ inside the circle given below, $AP$ and $CQ$ are the tangents to the circle at $A$ and $C$ respectively. If $\angle DAB:\angle DCB=11:7$, then the value of $\left(\angle BCQ+\angle DAP\right):{360}^0$ is

Prove Theorem of angle between tangent and secant for the following diagram

How can you prove the converse of the above theorem. "If a line in the plane of a circle is perpendicular to the radius at its end point on the circle, then the line is tangent to the circle " .

Prove that a parallelogram circumscribing a circle is a rhombus.

State and prove converse of alternate segment theorem

If line $AB$ is a secant of the circle where points $A,B$ are on the circle and the line $PQ$ is such that it touches the circle at point $A$, and $\angle BAQ=\angle ACB$ where point $C$ is in the corresponding alternative segment then line $PQ$ is not tangent to the circle at point $A$.

In the above diagram $PR$ is a secant, $PS$ is a tangent. The value of $PQ$ is $3 cm$, $QR$ is $9 cm$. The value of $PS$ is $3 cm$.

In the above diagram $PR$ is a secant, $PS$ is a tangent. The value of $PQ$ is $3 cm$, $QR$ is $2 cm$. Find the value of $PS$.

In the diagram, $BC$ is a chord and meets a diameter $AB$ at $B$. $AO=BO=3$ and $BC=8$. Find $x$ and $y$ .

Find the value of $x$ in the given diagram:

According to the converse of alternate segment theorem "If a line is drawn through an end point of a _____ of a circle so that the angle formed with the chord is equal to the angle subtended by the chord in the alternate segment, then the line is a tangent to the circle"

According to the converse of alternate segment theorem "If a line is drawn through an end point of a chord of a circle so that the angle formed with the chord is equal to the angle subtended by the chord in the alternate segment, then the line is a "

If $\angle ABC$ be an angle with the vertex $B$ that lies on a circle with the centre $O$. Line $BA$ is a secant of the circle that intersects the circle at the point $A$. Line $BC$ is a tangent at the point $B$. Point $E$ is the interior of the angle $\angle ABC$. Point $F$ is the exterior of the angle $\angle ABC$. Then $\angle ABC=$

If line $AB$ is a secant of the circle where points $A,B$ are on the circle and the line $PQ$ is such that it touches the circle at point $A$, and $\angle BAQ=$_____ where point $C$ is in the corresponding alternative segment then line $PQ$ is tangent to the circle at point $B$.

If line $AB$ is a secant of the circle where points $A,B$ are on the circle and the line $PQ$ is such that it touches the circle at point $A$, and $\angle BAQ=\angle ACB$ where point $C$ is in the corresponding alternative segment then

From an external point $P$, $PA$ and $PB$ are tangents to a circle with centre $O$. If $\angle POA=50°$, then the value of $\angle APB$ is 

Let $ABCD$ be a trapezium in which $AB\parallel CD$ and $AD\perp AB$. Suppose $ABCD$ has an incircle which touches $AB$ at $Q$ and $CD$ at $P$. Given that $PC=36$ and $QB=49,$ find $PQ.$

Suppose $S_1 \mathrm{a}\mathrm{n}\mathrm{d} S_2$ are two unequal circles; $AB$ and $CD$ are the direct common tangents to these circles. A transverse common tangent $PQ$ cuts $AB$ in $R$ and $CD$ in $S$. If $AB=10$ units, then $RS$ is -

Tangents to a circle at points $P$ and $Q$ on the circle intersect at a point $R$. If $PQ=6$ units and $PR=5$ units, then the radius of the circle is