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Tangent to A Circle: Definition and Examples | EDU.COM

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Tangent to A Circle: Definition and Examples | EDU.COMEDU.COMResourcesBlogGuidePodcastPlanBackHomesvg]:size-3.5">Math Glossarysvg]:size-3.5">Tangent to A CircleTangent to A Circle: Definition and ExamplesTable of ContentsTangent of a Circle Definition of Tangent to a Circle

A tangent of a circle is a straight line that touches the circle at only one point, known as the point of tangency. The word "tangent" comes from the Latin word "tangere," which means "to touch." This line never enters the interior of the circle but stays outside while making contact at exactly one point.

Tangents have several important properties. The tangent makes a right angle (90°) with the radius at the point of tangency. Two tangents drawn from an external point to a circle have equal lengths. A circle can have infinitely many tangents. From a point on the circle, exactly one tangent can be drawn, while from a point outside the circle, exactly two tangents can be drawn.

Examples of Tangent to a Circle Example 1: Finding the Length of a Tangent Using Pythagoras' Theorem Problem:

In the given figure, if AC=15AC = 15AC=15 inches and BC=25BC = 25BC=25 inches, find the length of the tangent.

Tangent To A Circle

Step-by-step solution:

Step 1, Look at what we know. We have AC=15AC = 15AC=15 inches and BC=25BC = 25BC=25 inches.

Step 2, Recall that the radius and tangent of a circle are perpendicular to each other at the point of tangency. This creates a right angle at point AAA.

Step 3, Use the Pythagoras' theorem in triangle ABCABCABC, which has a right angle at AAA:

AC2+AB2=BC2AC^2 + AB^2 = BC^2AC2+AB2=BC2

Step 4, Substitute the known values:

152+AB2=25215^2 + AB^2 = 25^2152+AB2=252

Step 5, Solve for $AB$:

AB2=252−152AB^2 = 25^2 - 15^2AB2=252−152

AB2=625−225AB^2 = 625 - 225AB2=625−225

AB2=400AB^2 = 400AB2=400

AB=400=20 inchesAB = \sqrt{400} =20 \text{ inches}AB=400​=20 inches

Example 2: Finding an Unknown Value Using Equal Tangent Lengths Problem:

Find the value of "x" in the figure given below.

Tangent To A Circle

Step-by-step solution:

Step 1, Recall the property that two tangent lines drawn from an external point to a circle have the same length. In this case, tangents PAPAPA and PBPBPB should be equal.

Step 2, Identify what we know about these tangents. We have PA=x2+10PA = x^2 + 10PA=x2+10 and PB=46PB = 46PB=46.

Step 3, Apply the equal tangent property:

PA=PBPA = PBPA=PB

x2+10=46x^2 + 10 = 46x2+10=46

Step 4, Solve for xxx:

x2=46−10x^2 = 46 - 10x2=46−10

x2=36x^2 = 36x2=36

x=6x = 6x=6

Example 3: Using the Tangent-Secant Formula Problem:

In the figure given below, if PA=9PA = 9PA=9 feet, AB=7AB = 7AB=7 feet, find the length of the tangent PQPQPQ.

Tangent To A Circle

Step-by-step solution:

Step 1, Identify what we know. We have PA=9PA = 9PA=9 feet and AB=7AB = 7AB=7 feet, where PAPAPA is part of a secant line.

Step 2, Calculate the total length of the secant from the external point:

PB=PA+AB=9 feet+7 feet=16 feetPB = PA + AB = 9 \text{ feet} + 7 \text{ feet} = 16 \text{ feet}PB=PA+AB=9 feet+7 feet=16 feet

Step 3, Apply the tangent-secant formula, which states that:

PQ2=PA×PBPQ^2 = PA \times PBPQ2=PA×PB

where PQPQPQ is the length of the tangent from point PPP.

Step 4, Substitute the known values:

PQ2=9×16PQ^2 = 9 \times 16PQ2=9×16

PQ2=144PQ^2 = 144PQ2=144

Step 5, Find the length of the tangent:

PQ=144=12 feetPQ = \sqrt{144} = 12 \text{ feet}PQ=144​=12 feet

Comments(2)RRunnerZachNovember 4, 2025This glossary def of tangent to a circle is great! It helped my students grasp the concept easily. Thanks for the clear explanation!

SSculptorOscarNovember 4, 2025I've used this tangent to a circle def for my students. It's clear & helped them grasp key concepts. Great resource!

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