# 12.06 Tangents to a circle

## Interactive practice questions

In this question we aim to prove that the tangent is perpendicular to the radius drawn from its point of contact.

In the diagram, $C$C is an arbitrary point on the line $AD$AD, and $B$B is the point at which the tangent meets the circle.

a

What can we say about the lines $OB$OB and $OC$OC?

$OB=OC$OB=OC

A

$OB>OC$OB>OC

B

$OBOB<OC C b What point on$AD$AD is closest to the center of the circle? Point$A$A A Point$B$B B Point$C$C C Point$D$D D c In general, what can we say about the angle of a line joining some point to some other line by the shortest route? The angle is obtuse. A The angle is reflex. B The angle is a right angle. C The angle is acute. D d Hence, what can we say about angle$\angle OBA$OBA? Straight angle A Acute angle B Reflex angle C Right angle D Easy Approx 2 minutes In the figure below,$AC$AC is tangent to both circles. In the diagram,$AC$AC is a tangent to the circle with center$O$O. What is the measure of$x$x? If$\overline{BA}$BA is a tangent to the circle, determine the value of$x\$x showing all steps of working.