EXAMPLE 2 Find an angle measure inside a circle Find the value of x. SOLUTION The chords JL and KM intersect inside the circle. Use Theorem 10.12. xoxo.

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Presentation transcript:

EXAMPLE 2 Find an angle measure inside a circle Find the value of x. SOLUTION The chords JL and KM intersect inside the circle. Use Theorem xoxo = 1212 (mJM + mLK) xoxo = 1212 (130 o o ) Substitute. xoxo = 143 Simplify.

EXAMPLE 3 Find an angle measure outside a circle Find the value of x. SOLUTION Use Theorem Substitute. Simplify. The tangent CD and the secant CB intersect outside the circle. = 1212 (178 o – 76 o ) xoxo = 51 x m BCD (mAD – mBD) = 1212

EXAMPLE 4 Solve a real-world problem SCIENCE 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 SOLUTION Use Theorem Substitute. 149 o 1212 [(360 o – x o ) –x o ] Solve for x. xoxo 31 = 1212 m BCD (mDEB – mBD) Because CB and CD are tangents, CB AB and CD AD Also,BC DC and CA CA. So, ABC ADC by the Hypotenuse-Leg Congruence Theorem, and BCA DCA. Solve right CBA to find that m BCA 74.5°. ANSWER The measure of the arc from which the flash is visible is about 31 o.

GUIDED PRACTICE for Examples 2, 3, and 4 4. Find the value of the variable. SOLUTION The chords AC and CD intersect inside the circle. Use Theorem Substitute. Simplify. = 1212 (y o + 95 o ) 78 o = y 61 78° (mAB + mCD) = 1212

GUIDED PRACTICE for Examples 2, 3, and 4 Find the value of the variable SOLUTION The tangent JF and the secant JG intersect outside the circle. Use Theorem Substitute. Simplify. = 1212 (a o – 44 o ) 30 o = 104 a m FJG (mFG – mKH) = 1212

GUIDED PRACTICE for Examples 2, 3, and 4 6. Find the value of the variable. SOLUTION Use Theorem Substitute o 1212 [(x o ) –(360 –x) o ] Solve for x. xoxo = 1212 m TQR (mTUR – mTR) Because QT and QR are tangents, QR RS and QT TS Also,TS SR and CA CA. So, QTS QRS by the Hypotenuse-Leg Congruence Theorem, and TQS RQS. Solve right QTS to find that m RQS 73.7°.