Questions: What is the measure of minor arc XZ, given that YX and YZ are tangent to circle O? A. 40° B. 180° C. 1900 D. 140°

Transcript text: What is the measure of minor arc $\overline{X Z}$, given that $\overline{Y X}$ and $\overline{Y Z}$ are tangent to $\odot 0$ ? A. $40 \circ$ B. $180 \circ$ C. 1900 D. $140 \circ$

Solution

Solution Steps

Step 1: Find the measure of major arc XZ

The sum of the measures of the arcs of a circle is 360°. The major arc XZ has a measure of 220°. The minor arc XZ has a measure of 360° - 220° = 140°.

Step 2: Find the measure of angle XYZ

The tangents YX and YZ form an angle whose measure is half the difference of the intercepted arcs' measures. Angle XYZ= (1/2)(major arc XZ - minor arc XZ) Angle XYZ= (1/2)(220°- minor arc XZ) Since the sum of the angles in a triangle is 180°, then 40°+angle XYZ + angle YZX=180° Since YX and YZ are tangents from the same external point Y, their lengths are equal, and triangle XYZ is isosceles, which means that the angles opposite the congruent sides are congruent. Thus, angle YZX = angle YXZ. Then, 40° + 2 * angle XYZ=180° 2* angle XYZ= 180° -40° 2* angle XYZ= 140° angle XYZ = 140/2 angle XYZ= 70°

Step 3: Set the equations equal to each other and solve for the minor arc XZ.

(1/2)(220° - minor arc XZ) = 70° 220° - minor arc XZ= 140° minor arc XZ= 220°- 140° minor arc XZ= 80°