Five-Minute Check (over Lesson 10–6)

CCSS

Then/Now

New Vocabulary

Theorem 10.15: Segments of Chords Theorem

Example 1:Use the Intersection of Two Chords

Example 2:Real-World Example: Find Measures of Segments in Circles

Theorem 10.16: Secant Segments Theorem

Example 3:Use the Intersection of Two Secants

Theorem 10.17

Example 4:Use the Intersection of a Secant and a Tangent

Over Lesson 10–6

A. 70

B. 75

C. 80

D. 85

Find x. Assume that any segment that appears to be tangent is tangent.

Over Lesson 10–6

A. 110

B. 115

C. 125

D. 130

Find x. Assume that any segment that appears to be tangent is tangent.

Over Lesson 10–6

A. 100

B. 110

C. 115

D. 120

Find x. Assume that any segment that appears to be tangent is tangent.

Over Lesson 10–6

A. 40

B. 38

C. 35

D. 31

Find x. Assume that any segment that appears to be tangent is tangent.

Over Lesson 10–6

A. 55

B. 110

C. 125

D. 250

What is the measure of XYZ if is tangent to the circle?

Content Standards

Reinforcement of G.C.4 Construct a tangent line from a point outside a given circle to the circle.

Mathematical Practices

1 Make sense of problems and persevere in solving them.

7 Look for and make use of structure.

You found measures of diagonals that intersect in the interior of a parallelogram.

• Find measures of segments that intersect in the interior of a circle.

• Find measures of segments that intersect in the exterior of a circle.

• chord segment

• secant segment

• external secant segment

• tangent segment

Use the Intersection of Two Chords

A. Find x.

AE • EC = BE • ED Theorem 10.15

x • 8 = 9 • 12 Substitution

8x = 108 Multiply.

x = 13.5 Divide each side by 8.

Answer: x = 13.5

Use the Intersection of Two Chords

B. Find x.

PT • TR = QT • TS Theorem 10.15

x • (x + 10) = (x + 2) • (x + 4) Substitution

x2 + 10x = x2 + 6x + 8 Multiply.

10x = 6x + 8 Subtract x2 from each side.

Use the Intersection of Two Chords

4x = 8 Subtract 6x from each side.

x = 2 Divide each side by 4.

Answer: x = 2

A. 12

B. 14

C. 16

D. 18

A. Find x.

A. 2

B. 4

C. 6

D. 8

B. Find x.

Find Measures of Segments in Circles

BIOLOGY Biologists often examine organisms under microscopes. The circle represents the field of view under the microscope with a diameter of 2 mm. Determine the length of the organism if it is located 0.25 mm from the bottom of the field of view. Round to the nearest hundredth.

Find Measures of Segments in Circles

Understand Two cords of a circle are shown. Youknow that the diameter is 2 mm and thatthe organism is 0.25 mm from thebottom.

Plan Draw a model using a circle. Let xrepresent the unknown measure of theequal lengths of the chordwhich is the length of the organism. Use the products of the lengths of the intersecting chords to findthe length of the organism.

Find Measures of Segments in Circles

Solve The measure of EB = 2.00 – 0.25 or1.75 mm.

HB ● BF = EB ● BG Segment products

x ● x = 1.75 ● 0.25 Substitution

x2 = 0.4375 Simplify.

x ≈ 0.66 Take the square root ofeach side.

Answer: The length of the organism is 0.66millimeters.

Find Measures of Segments in Circles

Check Use the Pythagorean Theorem to checkthe triangle in the circle formed by theradius, the chord, and part of thediameter.

1 ≈ 1

12 ≈ (0.75)2 + (0.66)2?

1 ≈ 0.56 + 0.44?

A. 10 ft

B. 20 ft

C. 36 ft

D. 18 ft

ARCHITECTURE Phil is installing a new window in an addition for a client’s home. The window is a rectangle with an arched top called an eyebrow. The diagram below shows the dimensions of the window. What is the radius of the circle containing the arc if the eyebrow portion of the window is not a semicircle?

Use the Intersection of Two Secants

Find x.

Use the Intersection of Two Secants

Answer: 34.5

Theorem 10.16

Substitution

Distributive Property

Subtract 64 from each side.

Divide each side by 8.

A. 28.125

B. 50

C. 26

D. 28

Find x.

Use the Intersection of a Secant and a Tangent

Answer: Since lengths cannot be negative, the value of x is 8.

LM is tangent to the circle. Find x. Round to the nearest tenth.

LM2 = LK ● LJ

122 = x(x + x + 2)

144 = 2x2 + 2x

72 = x2 + x

0 = x2 + x – 72

0 = (x – 8)(x + 9)

x = 8 or x = –9

A. 22.36

B. 25

C. 28

D. 30

Find x. Assume that segments that appear to be tangent are tangent.