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Segments from Secants and Tangents

Product of the outside segment and whole secant equals the square of the tangent to the same point.

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Segments from Secants and Tangents

Segments from Secants and Tangents 

If a tangent and secant meet at a common point outside a circle, the segments created have a similar relationship to that of two secant rays. Recall that the product of the outer portion of a secant and the whole is equal to the same of the other secant. If one of these segments is a tangent, it will still be the product of the outer portion and the whole. However, for a tangent line, the outer portion and the whole are equal.

Tangent Secant Segment Theorem: If a tangent and a secant are drawn from a common point outside the circle (and the segments are labeled like the picture to the left), then \begin{align*}a^2=b(b+c)\end{align*}a2=b(b+c). This means that the product of the outside segment of the secant and the whole is equal to the square of the tangent segment.

Finding the Value of a Missing Segment 

1. Find the value of the missing segment.

Use the Tangent Secant Segment Theorem. Square the tangent and set it equal to the outer part times the whole secant.

\begin{align*}x^2=4(4+12)\\ x^2=4 \cdot 16=64\\ x=8\end{align*}x2=4(4+12)x2=416=64x=8

2. Find the value of the missing segment.

Use the Tangent Secant Segment Theorem. Square the tangent and set it equal to the outer part times the whole secant.

\begin{align*}20^2=y(y+30)\\ 400=y^2+30y\\ 0=y^2+30y-400\\ 0=(y+40)(y-10)\\ y=\xcancel{-40},10\end{align*}202=y(y+30)400=y2+30y0=y2+30y4000=(y+40)(y10)y=40,10

Solving for Unknown Values 

Fill in the blank and then solve for the missing segment.

\begin{align*}\underline{\;\;\;\;\;\;\;}=\underline{\;\;\;\;\;\;\;}(4+5)\end{align*}=(4+5)

\begin{align*}x^2&=4(4+5)\\x^2&=36\\x&=6\end{align*}x2x2x=4(4+5)=36=6

Examples 

Find \begin{align*}x\end{align*}x in each diagram below. Use the Tangent Secant Segment Theorem and simplify any radicals.

Example 1

\begin{align*} 18^2&=10(10+x)\\324&=100+10x\\224&=10x\\x&=22.4\end{align*}182324224x=10(10+x)=100+10x=10x=22.4Example 2

\begin{align*}x^2&=16(16+25)\\x^2&=656\\x&=4\sqrt{41}\end{align*}x2x2x=16(16+25)=656=441Example 3

\begin{align*}x^2&=24(24+20)\\x^2&=1056\\x&=4\sqrt{66}\end{align*}x2x2x=24(24+20)=1056=466

Review 

Solve for the missing segment.

Find \begin{align*}x\end{align*}x in each diagram below. Simplify any radicals.

  1. Describe and correct the error in finding \begin{align*}y\end{align*}y. \begin{align*}10 \cdot 10&=y\cdot 15y\\ 100&=15y^2\\ \frac{20}{3}&=y^2\\ \frac{2\sqrt{15}}{3}&=y \quad {\color{red}\longleftarrow \ y} \ {\color{red}\text{is \underline{not} correct}}\end{align*}10101002032153=y15y=15y2=y2=y y is \underline{not} correct

Solve for the unknown variable.

  1. Find \begin{align*}x\end{align*}x and \begin{align*}y\end{align*}y.

Review (Answers)

To view the Review answers, open this PDF file and look for section 9.11. 

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Vocabulary

TermDefinition
central angle An angle formed by two radii and whose vertex is at the center of the circle.
chord A line segment whose endpoints are on a circle.
diameter A chord that passes through the center of the circle. The length of a diameter is two times the length of a radius.
inscribed angle An angle with its vertex on the circle and whose sides are chords.
intercepted arc The arc that is inside an inscribed angle and whose endpoints are on the angle.
point of tangency The point where the tangent line touches the circle.
Secant The secant of an angle in a right triangle is the value found by dividing length of the hypotenuse by the length of the side adjacent the given angle. The secant ratio is the reciprocal of the cosine ratio.
Tangents Secant Segments Theorem If a tangent and a secant are drawn from a common point outside the circle (and the segments are labeled like the picture below), then a^2 = b(b+c).

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