Special Segments in a Circle LESSON 10–7

Lesson Menu Five-Minute Check (over Lesson 10–6) TEKaS 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 Example 4:Use the Intersection of a Secant and a Tangent

Over Lesson 10–6 5-Minute Check 1 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 5-Minute Check 2 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 5-Minute Check 3 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 5-Minute Check 4 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 5-Minute Check 5 A.55 B.110 C.125 D.250 What is the measure of  XYZ if is tangent to the circle?

TEKS Targeted TEKS G.12(A) Apply theorems about circles, including relationships among angles, radii, chords, tangents, and secants, to solve non-contextual problems. Mathematical Processes G.1(B), G.1(E)

Then/Now 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.

Vocabulary chord segment secant segment external secant segment tangent segment

Concept

Example 1 Use the Intersection of Two Chords A. Find x. AE EC =BE EDTheorem x 8 =9 12Substitution 8x =108Multiply. x =13.5Divide each side by 8. Answer: x = 13.5

Example 1 Use the Intersection of Two Chords B. Find x. PT TR =QT TSTheorem x (x + 10) =(x + 2) (x + 4)Substitution x x =x 2 + 6x + 8Multiply. 10x =6x + 8Subtract x 2 from each side.

Example 1 Use the Intersection of Two Chords 4x =8Subtract 6x from each side. x =2Divide each side by 4. Answer: x = 2

Example 1 A.12 B.14 C.16 D.18 A. Find x.

Example 1 A.2 B.4 C.6 D.8 B. Find x.

Example 2 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.

Example 2 Find Measures of Segments in Circles AnalyzeTwo cords of a circle are shown. You know that the diameter is 2 mm and that the organism is 0.25 mm from the bottom. FormulateDraw a model using a circle. Let x represent the unknown measure of the equal lengths of the chord which is the length of the organism. Use the products of the lengths of the intersecting chords to find the length of the organism.

Example 2 Find Measures of Segments in Circles DetermineThe measure of EB = 2.00 – 0.25 or 1.75 mm. HB ● BF=EB ● BGSegment products x ● x=1.75 ● 0.25Substitution x 2 =0.4375Simplify. x≈0.66Take the square root of each side. Answer: The length of the organism is 0.66 millimeters.

Example 2 Find Measures of Segments in Circles JustifyUse the Pythagorean Theorem to check the triangle in the circle formed by the radius, the chord, and part of the diameter. 1≈11≈1 1 2 ≈(0.75) 2 + (0.66) 2 ? 1≈ ?

Example 2 Find Measures of Segments in Circles EvaluateWhen solving word problems involving circles, it is helpful to make a drawing and label all parts of the circle that are known. Use a variable to label the unknown measure.

Example 2 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?

Concept

Example 3 Use the Intersection of Two Secants Find x.

Example 3 Use the Intersection of Two Secants Answer: 34.5 Theorem Substitution Distributive Property Subtract 64 from each side. Divide each side by 8.

Example 3 A B.50 C.26 D.28 Find x.

Concept

Example 4 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. LM 2 =LK ● LJ 12 2 =x(x + x + 2) 144=2x 2 + 2x 72=x 2 + x 0=x 2 + x – 72 0=(x – 8)(x + 9) x=8 or x = –9

Example 4 A B.25 C.28 D.30 Find x. Assume that segments that appear to be tangent are tangent.

Special Segments in a Circle LESSON 10–7