10. Circles

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 $AD$`A``D`, and $B$`B` is the point at which the tangent meets the circle.

a

What can we say about the lines $OB$`O``B` and $OC$`O``C`?

$OB=OC$`O``B`=`O``C`

A

$OB>OC$`O``B`>`O``C`

B

$OB`O``B`<`O``C`

C

b

What point on $AD$`A``D` 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 $\angle OBA$∠`O``B``A`?

Straight angle

A

Acute angle

B

Reflex angle

C

Right angle

D

Easy

1min

In the figure below, $AC$`A``C` is tangent to both circles.

Easy

2min

In the diagram, $AC$`A``C` is a tangent to the circle with center $O$`O`. What is the measure of $x$`x`?

Easy

< 1min

If $\overline{BA}$`B``A` is a tangent to the circle, determine the value of $x$`x` showing all steps of working.

Easy

1min

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Apply geometric theorems to verify properties of circles.

Use chords, tangents, and secants to find missing arc measures or missing segment measures.