10. Circles

Lesson

A tangent is a line that intersects the circumference of a circle in exactly one point, which we call the point of tangency.

A tangent is perpendicular to the radius from the point of tangency. Conversely, the perpendicular to a radius through the same endpoint is a tangent line.

There can be more than one tangent on a circle. In fact there is basically an infinite number! The diagram below shows two tangents- $PM$`P``M` and $PQ$`P``Q`.

If two tangents are drawn from a common point, the tangents are equal.

Let's start by drawing in radii from the points of tangency:

In $\triangle OMP$△`O``M``P` and $\triangle OQP$△`O``Q``P`:

$OM=OQ$`O``M`=`O``Q`(radii in a circle are equal)

$OP$`O``P` is common

$\angle OMP=\angle OQP$∠`O``M``P`=∠`O``Q``P`$=$=$90^\circ$90° (tangents meet radii at right angles)

$\therefore$∴ $\triangle OMP$△`O``M``P`$\cong$≅$\triangle OQP$△`O``Q``P` (RHS)

$\therefore$∴ $MP=QP$`M``P`=`Q``P` (corresponding sides in congruent triangles are equal)

Remember, we aren't limited to the rules of circle geometry. We can use all our geometrical rules, including the Pythagorean theorem, congruency and similarity.

Remember!

The measure of an angle formed by a tangent and a chord is half the measure of the intercepted arc.

$\angle AED=\frac{1}{2}ABE$∠`A``E``D`=12`A``B``E`

If two chords intersect inside a circle, then the measure of each angle is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle:

$\angle1=\frac{1}{2}(x+y)$∠1=12(`x`+`y`)

The measure of the angle formed by the secants is half the difference between the measure of the intercepted arcs:

$\angle1=\frac{1}{2}\left(b-a\right)$∠1=12(`b`−`a`)

If $\angle FPB=\left(2x+17\right)^\circ$∠`F``P``B`=(2`x`+17)°, $AG=\left(3x+7\right)^\circ$`A``G`=(3`x`+7)°, and $FB=\left(2x-7\right)^\circ$`F``B`=(2`x`−7)°, solve for $x$`x`.

**Think**: By extending the chords $AB$`A``B` and $FG$`F``G` we get secants. We know the measure of two arcs and the measure of the angle between the two secants, so we can use the secant angle theorem to relate the quantities.

**Do**:

$\angle FPB$∠FPB |
$=$= | $\frac{1}{2}\left(AG+FB\right)$12(AG+FB) |

$2x+17$2x+17 |
$=$= | $\frac{1}{2}\left(\left(3x+7\right)+\left(2x-7\right)\right)$12((3x+7)+(2x−7)) |

$2\left(2x+17\right)$2(2x+17) |
$=$= | $5x$5x |

$4x+34$4x+34 |
$=$= | $5x$5x |

$x$x |
$=$= | $34^\circ$34° |

Given that $AC=78^\circ$`A``C`=78°, $AD=170^\circ$`A``D`=170°, and that $\overline{AB}$`A``B` is a tangent to the circle, find $\angle ABC$∠`A``B``C`.

**Think**: We know the measures of two arcs of the circle, and wish to find the measure of the angle outside of the circle formed by the corresponding secant and tangent. To find this, we can use the outside secant angle theorem.

**Do**:

$\angle ABC$∠ABC |
$=$= | $\frac{1}{2}\left(AD-AC\right)$12(AD−AC) |

$=$= | $\frac{1}{2}\left(170^\circ-78^\circ\right)$12(170°−78°) | |

$=$= | $46^\circ$46° | |

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.

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`CWhat point on $AD$

`A``D`is closest to the center of the circle?Point $A$

`A`APoint $B$

`B`BPoint $C$

`C`CPoint $D$

`D`DIn 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.

AThe angle is reflex.

BThe angle is a right angle.

CThe angle is acute.

DHence, what can we say about angle $\angle OBA$∠

`O``B``A`?Straight angle

AAcute angle

BReflex angle

CRight angle

D

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

Two tangents are drawn from an external point $B$`B` to the circle with center $O$`O`. What is the value of angle $x$`x`? Give reasons for your answer.