<meta http-equiv="refresh" content="1; url=/nojavascript/"> Segments from Secants and Tangents | CK-12 Foundation
Dismiss
Skip Navigation
You are reading an older version of this FlexBook® textbook: CK-12 Geometry Concepts Go to the latest version.

9.11: Segments from Secants and Tangents

Difficulty Level: At Grade Created by: CK-12
 0  0  0
%
Best Score
Practice Segments from Secants and Tangents
Practice
Best Score
%
Practice Now

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.

Watch This

CK-12 Foundation: Chapter9SegmentsfromSecantsandTangentsA

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 a^2=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.

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.

x^2=4(4+12)\\x^2=4 \cdot 16=64\\x=8

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.

20^2=y(y+30)\\400=y^2+30y\\0=y^2+30y-400\\0=(y+40)(y-10)\\y=\xcancel{-40},10

Example C

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

\underline{\;\;\;\;\;\;\;}=\underline{\;\;\;\;\;\;\;}(4+5)

x^2&=4(4+5)\\x^2&=36\\x&=6

Watch this video for help with the Examples above.

CK-12 Foundation: Chapter9SegmentsfromSecantsandTangentsB

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 x in each diagram below. Simplify any radicals.

1.

2.

3.

Answers:

Use the Tangent Secant Segment Theorem.

1.

 18^2&=10(10+x)\\324&=100+10x\\224&=10x\\x&=22.4

2.

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

3.

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

Practice

Solve for the missing segment.

Find x in each diagram below. Simplify any radicals.

  1. Describe and correct the error in finding y . 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}}

Solve for the unknown variable.

  1. Find x and y .

Image Attributions

Description

Difficulty Level:

At Grade

Grades:

Date Created:

Sep 24, 2012

Last Modified:

Aug 21, 2014
Files can only be attached to the latest version of Modality

Reviews

Please wait...
Please wait...
Image Detail
Sizes: Medium | Original
 
MAT.GEO.766.L.2
ShareThis Copy and Paste

Original text