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12.07 Lengths in chords, secants, and tangents

Lesson

Intersecting chords

An interesting phenomenon happens when two chords of the same circle intersect.

Exploration

Consider the applet below which shows two intersecting chords, $\overline{BC}$BC and $\overline{DE}$DE. Which two triangles in the diagram are similar? Why?

We can prove the triangles $\Delta BFD$ΔBFD and $\Delta EFC$ΔEFC in the applet are similar by angle-angle similarity and the fact that the angles intercept the same arcs. Using the similar triangles, we have the following relationship between the segment lengths.

$\triangle BFD$BFD $\sim$~ $\triangle EFC$EFC

Angle-angle similarity

$\frac{DF}{CF}$DFCF $=$= $\frac{BF}{EF}$BFEF

If triangles are similar, corresponding side lengths are proportional.

$DF\times EF$DF×EF $=$= $CF\times BF$CF×BF

Cross multiply

 

This relationship between the chord lengths is known as the intersecting chord theorem, and we can use it to solve for unknown lengths in the chords of a circle.

Intersecting chord theorem

If two chords intersect each other inside a circle, the products of their segment lengths are equal.

For example, in the circle below the segment lengths denoted by $a$a, $b$b, $c$c, and $d$d have the relationship $ab=cd$ab=cd.

In the chords above, $ab=cd$ab=cd

 

Practice question

Question 1

Solve for $x$x.

 

 

Intersecting tangents and secants

Now let's explore what happens to the lengths of two secants that intersect outside the circle.

Exploration

Similar triangles are also formed when two secants intersect outside the circle. Drag the points $A$A, $C$C, and $P$P around until $\overline{CP}$CP and $\overline{AP}$AP form a pair of intersecting secants. Then, use the double arrows to walk through the steps.

Why is $\triangle CPA$CPA similar to $\triangle BPD$BPD?

Let's explain what's happening in each step in the case where $ABDC$ABDC forms an inscribed quadrilateral:

1 $\overline{AB}$AB and $\overline{CD}$CD are secants of circle $O$O

Given

2 Draw $\overline{CA}$CAand $\overline{DB}$DB

Between any two points, there exists a line.

3 $\angle P$P is congruent to itself

Reflexive property of congruence.

4 $\angle CDB$CDB and $\angle BAC$BAC are supplementary

Opposite angles in an inscribed quadrilateral are supplementary.

  $\angle CDB$CDB and $\angle BDP$BDP are supplementary

Linear pairs are supplementary.

5 $\angle BAC\cong\angle BDP$BACBDP

Congruent supplements theorem.

 

Therefore, the two triangles are congruent by angle-angle similarity.

Now, since we have congruent triangles, we can relate the sides in a proportion (as we did with the lengths of chords). This gives us another theorem relating the lengths of intersecting secants.

Intersecting secants theorem

If two secant segments intersect outside the circle at a point, then the product of the measures of one secant segment and its external secant segment is equal to the product of the measures of the other secant segment and its external secant segment.

For example, in the diagram below we have the relationship

$a\left(a+b\right)=c\left(c+d\right)$a(a+b)=c(c+d)

Two secants meet at an external point.

 

Practice question

Question 2

Find the value of $x$x.

 

If we move the point $A$A in the applet above so that it coincides with point $B$B, then $\overline{AP}$AP becomes tangent to circle $O$O at point $A$A. secant and tangent. The same segment relationships apply to an intersecting tangent-secant pair as two secants.

Intersecting tangent-secant theorem

If a tangent segment and a secant segment are drawn to a circle from an exterior point, then the square of the measure of the tangent segment is equal to the product of the measures of the secant segment and its external secant segment.

For example, in the diagram below we have the relationship

$a^2=b\left(b+c\right)$a2=b(b+c)

A secant and a tangent meet at an external point.

Similarity if we move point $B$B so that it coincides with point $D$D, we create two tangent segments that meet outside the circle. By algebraic manipulation, we can show that the lengths of the two tangent segments are equal. This gives us the intersecting tangents theorem discussed previously.

Intersecting tangents theorem

If two tangent segments intersect outside the circle at the same point, then they are congruent.

Practice question

Question 3 

Find the value of $x$x.

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