2 Marks Questions

Question 12 Marks

In Figure, if TP and TQ are two tangents to a circle with centre O so that $\angle POQ = {110^o}$, then $\angle$PTQ is equal to:

Answer

Steps of Construction:

Draw a circle with any radius and center O. Here xy is given line.
Choose any point P on the circumference of the circle, and draw a line passing through P, Let's name it AB.
Draw a line AB parallel to xy, such that AB intersects the circle at two points P and A.Here, AB and xy are two parallel lines. AB intersects the circle at exactly two points, P and Q. Therefore, line AB is the secant of this circle.
CD intersects the circle at exactly one point, R, line CD is the tangent to the circle.

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Question 22 Marks

Two concentric circle are of radil 5 cm and 3 cm. find the length of the chord of the larger circle which touches the smaller circle.

Answer

Let the two concentric circles be centered at point $O$. And let $P Q$ be the chord of the larger circle which touches the smaller circle at point A . Therefore, PQ is tangent to the smaller circle.
$OA \perp PQ \quad . . .( As$ $OA$ is the radius of the circle)
Applying Pythagoras theorem in $\triangle OAP$, we obtain
$OA^2+AP^2=OP^2$
$3^2+AP^2=5^2$
$9+AP^2=25$
$AP^2=16$
$AP=4$
$\text { In } \triangle OPQ$
$\text { Since } OA \perp PQ$
$AP = AQ \quad \ldots$ (Perpendicular from the center of the circle bisects the chord)
$\therefore P Q=2 A P$
$=2 \times 4$
$=8$
Therefore, the length of the chord of the larger circle is 8 cm.

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