(A) The radius of the circle is $OP = 9$. Since $PQ$ is a tangent to the circle at point $P$,the radius $OP$ is perpendicular to the tangent $PQ$ at t

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(A) The radius of the circle is $OP = 9$. Since $PQ$ is a tangent to the circle at point $P$,the radius $OP$ is perpendicular to the tangent $PQ$ at the point of contact. Therefore,$\angle OPQ = 90^{\circ}$. In the right-angled triangle $\triangle OPQ$,by the Pythagorean theorem,we have: $OQ^2 = OP^2 + PQ^2$. Substituting the given values: $OQ^2 = 9^2 + 40^2$. $OQ^2 = 81 + 1600$. $OQ^2 = 1681$. Taking the square root of both sides: $OQ = \sqrt{1681} = 41$. Thus,the length of $OQ$ is $41$.

Class 10 Mathematics Circles Mix Examples - Circles

Medium

$A$ line passing through the centre of the circle $O$ intersects the tangent to the circle at $Q$. $P$ is the point of contact. If the radius of the circle is $9$ and $PQ = 40$,then find $OQ$.

A
$41$
B
$48$
C
$55$
D
$31$

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Prove that the tangent drawn at the mid-point of an arc of a circle is parallel to the chord joining the end points of the arc.

Difficult

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$A$ tangent line drawn from a point $A$ outside $\odot(P, 5)$ touches the circle at $B.$ If $PA = 13$,then $AB = \dots$

Medium

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$P$ is a point in the exterior of $\odot(O, r)$ and the tangents from $P$ to the circle touch the circle at $X$ and $Y$. Find $m\angle XOP$,if $m\angle XPO = 65^\circ$. (in $^\circ$)

Medium

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Which of the following groups correctly matches the data of Part $I$ with the data of Part $II$?

Part $I$ Part $II$

$1.$ In $\Delta ABC, AB=3, BC=4, AC=5$ $a.$ In-radius $= 1$

$2.$ In $\Delta PQR, PQ=5, QR=12, PR=13$ $b.$ In-radius $= 2$

$3.$ In $\Delta XYZ, XY=8, YZ=15, XZ=17$ $c.$ In-radius $= 3$

$4.$ In $\Delta MNP, MN=20, NP=21, MP=29$ $d.$ In-radius $= 6$

Difficult

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In a right triangle $ABC$ in which $\angle B = 90^{\circ}$,a circle is drawn with $AB$ as diameter,intersecting the hypotenuse $AC$ at $P$. Prove that the tangent to the circle at $P$ bisects $BC$.

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Part $I$	Part $II$
$1.$ In $\Delta ABC, AB=3, BC=4, AC=5$	$a.$ In-radius $= 1$
$2.$ In $\Delta PQR, PQ=5, QR=12, PR=13$	$b.$ In-radius $= 2$
$3.$ In $\Delta XYZ, XY=8, YZ=15, XZ=17$	$c.$ In-radius $= 3$
$4.$ In $\Delta MNP, MN=20, NP=21, MP=29$	$d.$ In-radius $= 6$

$A$ line passing through the centre of the circle $O$ intersects the tangent to the circle at $Q$. $P$ is the point of contact. If the radius of the circle is $9$ and $PQ = 40$,then find $OQ$.

Similar Questions

Prove that the tangent drawn at the mid-point of an arc of a circle is parallel to the chord joining the end points of the arc.

$A$ tangent line drawn from a point $A$ outside $\odot(P, 5)$ touches the circle at $B.$ If $PA = 13$,then $AB = \dots$

$P$ is a point in the exterior of $\odot(O, r)$ and the tangents from $P$ to the circle touch the circle at $X$ and $Y$. Find $m\angle XOP$,if $m\angle XPO = 65^\circ$. (in $^\circ$)

In a right triangle $ABC$ in which $\angle B = 90^{\circ}$,a circle is drawn with $AB$ as diameter,intersecting the hypotenuse $AC$ at $P$. Prove that the tangent to the circle at $P$ bisects $BC$.

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