Geometry (B): READ: Segments of Chords, Secants, and Tangents

Segments of Chords, Secants, and Tangents

Learning Objectives

Find the lengths of segments associated with circles.

In this section we will discuss segments associated with circles and the angles formed by these segments. The figures below give the names of segments associated with circles.

Segments of Chords

Theorem If two chords intersect inside the circle so that one is divided into segments of length a and b and the other into segments of length b and c then the segments of the chords satisfy the following relationship: $ab = cd.$

This means that the product of the segments of one chord equals the product of segments of the second chord.

Proof

We connect points $A$ and $C$ and points $D$ and $B$ to make $\triangle AEC$ and $\triangle DEB$ .

$& \angle{AEC} \cong \angle{DEB} && \text{Vertical angles}\\ & \angle{CAB} \cong \angle{BDC} && \text{Inscribed angles intercepting the same arc}\\ & \angle{ACD} \cong \angle{ABD} && \text{Inscribed angles intercepting the same arc}$

Therefore, $\triangle AEC \sim \triangle DEB$ by the AA similarity postulate.

In similar triangles the ratios of corresponding sides are equal.

$\frac{c} {b} = \frac{a} {d} \Rightarrow ab = cd \blacklozenge$

Example 1

Find the value of the variable.

$10x &= 8\times{12}\\ 10x &= 96\\ x &= 9.6$

Segments of Secants

Theorem If two secants are drawn from a common point outside a circle and the segments are labeled as below, then the segments of the secants satisfy the following relationship:

$a (a + b) = c (c + d)$

This means that the product of the outside segment of one secant and its whole length equals the product of the outside segment of the other secant and its whole length.

Proof

We connect points $A$ and $D$ and points $B$ and $C$ to make $\triangle BCN$ and $\triangle ADN$ .

$& \angle{BNC} \cong \angle{DNA} && \text{Same angle}\\ & \angle{NBC} \cong \angle{NDA}&& \text{Inscribed angles intercepting the same arc}$

Therefore, $\triangle BCN \sim \triangle DAN$ by the AA similarity postulate.

In similar triangles the ratios of corresponding sides are equal.

$\frac{a} {c} = \frac{c+d} {a+b} \Rightarrow a(a+b) = c(c+d) \blacklozenge$

Example 2

Find the value of the variable.

$10(10 + x) &= 9(9 + 20)\\ 100 + 10x &= 261\\ 10x &= 161\\ x &= 16.1$

Segments of Secants and Tangents

Theorem If a tangent and a secant are drawn from a point outside the circle then the segments of the secant and the tangent satisfy the following relationship

$a (a + b) = c^2.$

This means that the product of the outside segment of the secant and its whole length equals the square of the tangent segment.

Proof

We connect points $C$ and $A$ and points $B$ and $C$ to make $\triangle BCD$ and $\triangle CAD$ .

$m\angle{CDB}& =m\angle{BAC}-m\angle{DBC} && \text{The measure of an Angle outside a circle is equal to half} \\ &&& \text{the difference of the measures of the intercepted arcs}\\ m\angle{BAC}& =m\angle{ACD}+m\angle{CDB} && \text{The measure of an exterior angle in a triangle equals}\\ &&& \text{the sum of the measures of the remote interior angles}\\ m\angle{CDB}& =m\angle{ACD}+m\angle{CDB}-m\angle{DBC} && \text{Combining the two steps above}\\ m\angle{DBC}& =m\angle{ACD} && \text{algebra}$

Therefore, $\triangle BCD \sim \triangle CAD$ by the AA similarity postulate.

In similar triangles the ratios of corresponding sides are equal.

$\frac{c} {a + b} = \frac{a} {c} \Rightarrow a(a + b) = c^2 \blacklozenge$

Example 3

Find the value of the variable $x$ assuming that it represents the length of a tangent segment.

$x^2 &= 3(9+3)\\ x^2 &= 36\\ x &= 6$

Lesson Summary

In this section, we learned how to find the lengths of different segments associated with circles: chords, secants, and tangents. We looked at cases in which the segments intersect inside the circle, outside the circle, or where one is tangent to the circle. There are different equations to find the segment lengths, relating to different situations.

The questions are for your own review. The answers are listed below to help you check your work and understanding.

Review Questions

Find the value of missing variables in the following figures:
A circle goes through the points $A, B, C,$ and $D$ consecutively. The chords $\overline{AC}$ and $\overline{BD}$ intersect at $P$ . Given that $AP = 12, BP = 16,$ and $CP = 6,$ find $DP$ ?
Suzie found a piece of a broken plate. She places a ruler across two points on the rim, and the length of the chord is found to be $6\;\mathrm{inches.}$ The distance from the midpoint of this chord to the nearest point on the rim is found to be $1\;\mathrm{inch.}$ Find the diameter of the plate.
Chords $\overline{AB}$ and $\overline{CD}$ intersect at $P$ . Given $AP = 12, BP = 8,$ and $CP = 7,$ find $DP$ .

Review Answers

1. $14.4$
2. $16$
3. $4.5$
4. $32.2$
5. $12$
6. $29.67$
7. $4.4$
8. $18.03$
9. $4.54$
10. $20.25$
11. $7.48$
12. $23.8$
13. $24.4$
14. $9.24$ or $4.33$
15. $17.14$
16. $26.15$
17. $7.04$
18. $9.8$
19. $4.4$
20. $8$
$4.5$
$10\;\mathrm{inches.}$
$13.71$

Last modified: Tuesday, June 29, 2010, 11:39 AM