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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

What if you were given a circle with a tangent and a secant that intersect outside the circle? How could you use the length of some of the segments formed by their intersection to determine the lengths of the unknown segments? After completing this Concept, you'll be able to use the Tangent Secant Segment Theorem to solve problems like this one.

Guidance

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*}. This means that the product of the outside segment of the secant and the whole is equal to the square of the tangent segment.

Example A

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*}

Example B

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*}

Example C

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

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

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

Watch this video for help with the Examples above.

Vocabulary

A circle is the set of all points that are the same distance away from a specific point, called the center. A radius is the distance from the center to the circle. A chord is a line segment whose endpoints are on a circle. A diameter is a chord that passes through the center of the circle. The length of a diameter is two times the length of a radius. A central angle is the angle formed by two radii and whose vertex is at the center of the circle. An inscribed angle is an angle with its vertex on the circle and whose sides are chords. The intercepted arc is the arc that is inside the inscribed angle and whose endpoints are on the angle. A tangent is a line that intersects a circle in exactly one point. The point of tangency is the point where the tangent line touches the circle. A secant is a line that intersects a circle in two points.

Guided Practice

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

1.

2.

3.

Answers:

Use the Tangent Secant Segment Theorem.

1.

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

2.

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

3.

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

Practice

Solve for the missing segment.

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

1. Describe and correct the error in finding \begin{align*}y\end{align*}. \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*}

Solve for the unknown variable.

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

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Vocabulary Language: English Spanish

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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