Answer:
The measure of arc CD is 114.6°.
Step-by-step explanation:
Let the center of the circle be O and OP is perpendicular to CD at point point P.
The perpendicular line from the center of circle divide each chord in two equal parts. so, CP=DP.
[tex]CP=\frac{CD}{2}=\frac{36.7}{2}=18.35[/tex]
In a right angles triangle,
[tex]\sin\theta =\frac{opposite}{hypotenuse}[/tex]
In a right angles triangle OPC,
[tex]\sin\angle POC =\frac{18.35}{21.8}[/tex]
[tex]\angle POC =\sin^{-1} (\frac{18.35}{21.8})[/tex]
[tex]\angle POC \approx 57.3[/tex]
In triangle OPC and OPD,
[tex]OC=OD[/tex] (Radius)
[tex]OP=OP[/tex] (Reflexive property)
[tex]CP=DP[/tex] (Altitude on chord from center)
By SSS postulate
[tex]\triangle OPC\cong \triangle OPD[/tex]
[tex]\angle POC\cong \angle POD[/tex] (CPCTC)
[tex]\angle POC=\angle POD[/tex]
The measure of angle COD is
[tex]\angle COD=\angle POC+\angle POD[/tex]
[tex]\angle COD=2\angle POC[/tex]
[tex]\angle COD=2(57.3)[/tex]
[tex]\angle COD=114.6[/tex]
Therefore the measure of arc CD is 114.6°.
