1 / 7

Understanding Angles and Chords in Circles: Theorems and Real-World Applications

This guide explores the relationships of angles and chords in circles through examples and guided practice. It includes solving for angle measures both inside and outside circles using Theorem 10.12 and Theorem 10.13, while addressing real-world applications such as the visibility of the Northern Lights from Earth's surface. Detailed solutions illustrate substituting and simplifying expressions to find unknown variables. Whether you are enhancing your geometry skills or applying these concepts to practical problems, this resource will help clarify important theorems and their uses.

dima
Télécharger la présentation

Understanding Angles and Chords in Circles: Theorems and Real-World Applications

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. The chords JLand KMintersect inside the circle. (mJM + mLK) xo = 12 12 xo (130o + 156o) = xo = 143 EXAMPLE 2 Find an angle measure inside a circle Find the value of x. SOLUTION Use Theorem 10.12. Substitute. Simplify.

  2. The tangent CDand the secant CBintersect outside the circle. (mAD – mBD) m BCD = 12 12 xo (178o – 76o) = x = 51 EXAMPLE 3 Find an angle measure outside a circle Find the value of x. SOLUTION Use Theorem 10.13. Substitute. Simplify.

  3. The Northern Lights are bright flashes of colored light between 50 and 200 miles above Earth. Suppose a flash occurs 150 miles above Earth. What is the measure of arc BD, the portion of Earth from which the flash is visible? (Earth’s radius is approximately 4000 miles.) EXAMPLE 4 Solve a real-world problem SCIENCE

  4. Because CBand CDare tangents, Also,BC DC and CA CA . So, ABC ADC by the Hypotenuse-Leg Congruence Theorem, and BCA DCA.Solve rightCBA to find thatm BCA 74.5°. (mDEB – mBD) m BCD = CB AB and CD AD 12 12 149o [(360o – xo) –xo] xo 31 The measure of the arc from which the flash is visible is about 31o. ANSWER EXAMPLE 4 Solve a real-world problem SOLUTION Use Theorem 10.13. Substitute. Solve for x.

  5. The chords ACand CDintersect inside the circle. = 12 12 78o (yo + 95o) = 61 = y for Examples 2, 3, and 4 GUIDED PRACTICE 4. Find the value of the variable. SOLUTION (mAB + mCD) 78° Use Theorem 10.12. Substitute. Simplify.

  6. SOLUTION The tangent JFand the secant JGintersect outside the circle. (mFG – mKH) m FJG = 12 12 30o (ao – 44o) = a = 104 for Examples 2, 3, and 4 GUIDED PRACTICE Find the value of the variable. 5. Use Theorem 10.13. Substitute. Simplify.

  7. Because QTand QRare tangents, Also,TS SR and CA CA . So, QTS QRS by the Hypotenuse-Leg Congruence Theorem, and (mTUR – mTR) m TQR QR RS and QT TS 12 12 TQS RQS.Solve rightQTS to find thatm RQS 73.7°. = 73.7o [(xo) –(360 –x)o] xo 253.7 for Examples 2, 3, and 4 GUIDED PRACTICE 6. Find the value of the variable. SOLUTION Use Theorem 10.13. Substitute. Solve for x.

More Related