10.5 and 10.6 Secant and Tangent Lines

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Tangents and Secants

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VOCAB:

Secant is any line or ray that intersects a circle in exactly
two points.

Warm up:

10.5 & 10.6 Tangents and Secants

Multiply:

1.

𝑥 + 2 ²

2.

𝑥 − 3 ²

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Tangents and Secants

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Florida’s B.E.S.T. Standards for
Mathematics
MA.912.GR.6.1

Solve mathematical and real-world problems
involving the length of a secant, tangent,
segment or chord in a given circle.

MA.912.GR.6.2

Solve mathematical and real-world problems
involving the measures of arcs and related
angles.

MA.K12.MTR.4.1
Engage in discussions that reflect on the
mathematical thinking of self and others.

MA.K12.MTR.6.1
Assess the reasonableness of solutions.

Florida’s Mathematical Thinking
and Reasoning Standards

Lesson Objectives
Content Objective

Students solve problems using relationships between circles, tangents, and secants.

Language Objectives

•Students talk about tangents and secants using represent.

•To maximize linguistic and cognitive meta-awareness, students participate in MLR2: Collect and Display.

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Tangents and Secants

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Learn
Tangents, Secants, and Angle Measures

A secantis any line or ray that intersects a
circle in exactly two points. Lines j and k are
secants of ⊙C.

When two secants intersect inside a circle, the
angles formed are related to the arcs they
intercept.

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Tangents and Secants

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On the Circle

𝑚∠1 = 𝑥

2

Outside the Circle

Inside the Circle

Learn
Intercepted Arcs

𝑚∠1 = (𝑥 − 𝑦)

2

𝑚∠1 = (𝑥 + 𝑦)

2

x

1

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Tangents and Secants

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Example 1
Intersecting Chords or Secants

Find the value of x.

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Tangents and Secants

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Check 1
Arcs and Chords

Check

Find m 𝐺𝐻

Find m∠R

Multiple Choice

Check 1 -

Find measure of arc GH

• 45

• 14

• 22.5

• 36.5

Multiple Choice

Find the measure of angle R

• 60

• 40

• 80

• 20

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Tangents and Secants

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Learn
Tangents

A tangent to a circle is a line
or segment in the plane of a
circle that intersects the circle
in exactly one point and does
not contain any points in the
interior of the circle.

For a line that intersects a
circle in one point, the point of
tangency is the point where
they intersect.

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Tangents and Secants

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Learn
Tangents

A common tangentis a line
or segment that is tangent to
two circles in the same plane.

More than one line can be
tangent to the same circle.

The shortest distance from a
tangent to the center of a
circle is the radius drawn to
the point of tangency.

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Tangents and Secants

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Learn
Tangents

Theorem 10.11

In a plane, a line is tangent to a circle if and only if it is
perpendicular to a radius drawn to the point of tangency.

Theorem 10.12: Tangent to a Circle Theorem

If two segments from the same exterior point are tangent
to a circle, then they are congruent.

Theorem 10.13
If two segments or lines are tangent to a circle, then the
circumscribed angle and the central angle that intercept
the arc formed by the points of tangency are
supplementary.

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Tangents and Secants

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Example 2
Identify a Tangent

𝑨𝑩 is a radius of ⊙𝑨. Determine
whether 𝑩𝑪 is tangent to ⊙𝑨. Justify
your answer.

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Tangents and Secants

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Example 3
Use a Tangent to Find Missing Values

𝑸𝑺 is tangent to ⊙𝑹 at Q. Find the
value of x.

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Tangents and Secants

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Example 3
Use a Tangent to Find Missing Values

Check

𝐵𝐶 is tangent to ⊙𝐴 at C. Find the
value of x to the nearest hundredth.

Multiple Choice

Find the value of x to the nearest hundredth.

• 9.6

• 3

• 9.64

• 11.21

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Tangents and Secants

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Example 4
Use Congruent Tangents to Find Measures

PHOTOGRAPHY A photographer wants
to take a picture of a local fountain. She
positions herself at point A so that the
fountain will be centered in the picture.
𝑨𝑩and 𝑨𝑪are tangent to the fountain as
shown. If the lengths of the tangents are
given in feet, find AB.

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Tangents and Secants

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Example 4
Use Congruent Tangents to Find Measures

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Example 4
Find Measures in Circumscribed Polygons

△JKL is circumscribed about ⊙Q. Find the
perimeter of △JKL.

Draw

Label KM, LN, and JP with the correct lengths.

Multiple Choice

Triangle JKL is circumscribed about Circle Q. Find the perimeter of Triangle JKL.

• 32

• 64

• 360

• 180

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Tangents and Secants

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Example 5
Use Circumscribed Angles to Find Measures

If 𝒎∠𝑬𝑮𝑭 = (𝟏𝟗𝒙 + 𝟗) ° 𝐚𝐧𝐝
𝒎∠𝑫 = (𝟏𝟎𝒙 − 𝟑) °, find m∠D.

​

Poll

How confident do you feel about this topic (secants and tangent lines)?

• Not at all confident - please help/reteach this

• Somewhat confident - I still have a few questions

• Pretty confident -

  I just need to practice more

• Very confident -

  I could do well on a quiz/test about this

Complete the back WS - to be turned in at the end of class.
You may choose either EVENS (2,4,6, etc.) or ODDS (1,3,5, etc.) to complete

​

​Time to practice