Answer:
The measure of the third arc is [tex]150\°[/tex]
Step-by-step explanation:
step 1
we know that
The measurement of the external angle is the semi-difference of the arcs which comprises
in this problem
Let
x----> the greater arc of the circle intercepted by the secant and the tangent
y----> the smaller arc of the circle intercepted by the secant and the tangent
[tex]42\°=\frac{1}{2}(x-y)[/tex]
[tex]84\°=(x-y)[/tex] ----> equation A
[tex]\frac{x}{y}=\frac{7}{3}[/tex]
[tex]x=\frac{7}{3}y[/tex] -----> equation B
Substitute equation B in equation A and solve for y
[tex]84\°=(\frac{7}{3}y-y)[/tex]
[tex]84\°=(\frac{4}{3}y)[/tex]
[tex]y=3*84\°/4=63\°[/tex]
Find the value of x
[tex]x=\frac{7}{3}(63\°)=147\°[/tex]
step 2
Find the measure of the third arc
Let
z------> the measure of the third arc
we know that
[tex]x+y+z=360\°[/tex] -----> complete circle
substitute the values and solve for z
[tex]147\°+63\°+z=360\°[/tex]
[tex]z=360\°-(147\°+63\°)=150\°[/tex]
