9.7: Segments of Secants and Tangents
Learning Objectives
- Find the lengths of segments of secants and tangents.
Secant and Tangent Segments
In this section we will discuss segments (or parts of lines) associated with circles, and the angles formed by these segments. The figures below give the names of segments associated with circles:
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:
\begin{align*}a(a+b) = c(c+d)\end{align*}
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.
- Multiply the outside part of one _______________________ by the whole length of that secant, and it will equal the product of the outside part of the other secant and its whole length.
Proof
We connect points \begin{align*}A\end{align*} and \begin{align*}D\end{align*} and points \begin{align*}B\end{align*} and \begin{align*}C\end{align*} to make \begin{align*}\Delta BCN\end{align*} and \begin{align*}\Delta ADN\end{align*}.
Statements | Reasons |
---|---|
1. \begin{align*}\angle BNC \cong \angle DNA\end{align*} | 1. These are the same angle. |
2. \begin{align*}\angle NBC \cong \angle NDA\end{align*} | 2. Both inscribed angles intercept the same arc, so the angles are congruent. |
3. \begin{align*}\Delta BCN \sim \Delta DAN\end{align*} | 3. AA Similarity Postulate |
4. \begin{align*}\frac{a}{c} = \frac{c + d}{a + b}\end{align*} | 4. In similar triangles, the ratios of corresponding sides are equal. |
5. \begin{align*}a(a+b) = c(c+d)\end{align*} | 5. Cross multiplication |
Example 1
Find the value of the variable \begin{align*}x\end{align*}:
Use the product of secant segments:
\begin{align*}10(10 + x) & = 9(9 +20)\\ 100 + 10x & = 9(29)\\ 100 + 10x & = 261\\ 10x & = 161\\ x & = 16.1\end{align*}
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:
\begin{align*}a(a+b) = c^2\end{align*}
This means that the product of the outside segment of the secant and its whole length equals the square of the tangent segment.
- The __________________________ segment squared is the same as the product of the outside part of the secant and the secant’s whole length.
Before we prove this theorem, let’s review the two types of segment relationships you just learned. Complete the following table:
Segments | Draw a picture | Relationship |
---|---|---|
Secant – Secant (intersection outside of a circle) | ||
Tangent – Secant (intersection outside of a circle) |
Proof
We connect points \begin{align*}C\end{align*} and \begin{align*}A\end{align*} and points \begin{align*}B\end{align*} and \begin{align*}C\end{align*} to make \begin{align*}\Delta BCD\end{align*} and \begin{align*}\Delta CAD\end{align*}.
Statements | Reasons |
---|---|
1. \begin{align*}m \angle CDB = m \angle BAC - m \angle DBC\end{align*} | 1. The measure of an angle outside a circle is equal to half the difference of the measures of the intercepted arcs (or their corresponding angles). |
2. \begin{align*}m \angle BAC = m \angle ACD + m \angle CDB\end{align*} | 2. The measure of an exterior angle in a triangle is equal to the sum of the measures of the remote interior angles. |
3. \begin{align*}m \angle CDB = m \angle ACD + m \angle CDB - m \angle DBC\end{align*} | 3. Substitution |
4. \begin{align*}m \angle DBC = m \angle ACD\end{align*} | 4. Subtract and simplify |
5. \begin{align*}\Delta BCD \sim \Delta CAD\end{align*} | 5. AA Similarity Postulate |
6. \begin{align*}\frac{c}{a + b} = \frac{a}{c}\end{align*} | 6. In similar triangles, the ratios of corresponding sides are equal. |
7. \begin{align*}a(a+b) = c^2\end{align*} | 7. Cross multiplication |
This proof reviewed some postulates and theorems that you learned earlier in this unit and some that you learned in past units:
- The measure of an angle outside a circle is equal to _________________ the difference of the measures of the intercepted arcs, which are the same as the arcs’ corresponding angles.
- The measure of an _________________________ angle in a triangle is equal to the sum of the measures of the remote interior angles.
- The AA Similarity Postulate says that two triangles are ____________________ if they have two congruent corresponding angles.
Example 2
Find the value of the variable \begin{align*}x\end{align*} assuming that it represents the length of a tangent segment:
The tangent segment squared is equal to the product of the secant segments:
\begin{align*}x^2 & = 3(9 + 3)\\ x^2 & = 3(12) = 36\\ x & = 6\end{align*}
Reading Check:
In the space below, make up a problem that involves two segments that intersect outside of a circle. Your problem can use two secants or a secant and a tangent.
Draw a clear picture and label each segment. Make sure to include a variable for the missing segment. Then, solve your problem.
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
Graphic Organizer for Unit 9
Circle Part(s) | Draw a Picture | What is it? | Is there a relationship I need to know? | |
---|---|---|---|---|
Inscribed Angle | ||||
Intercepted Arc | ||||
Central Angle | ||||
2 Chords that intersect inside a circle | Angles/Arcs | Segments | ||
Tangent | ||||
2 Secants that intersect outside a circle | Angles/Arcs | Segments | ||
2 Tangents that intersect outside a circle | ||||
A Tangent and a Secant that intersect outside a circle | Angles/Arcs | Segments |
Image Attributions
Concept Nodes:
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