Question

A secant and a tangent to a circle intersect in a 42-degree angle. The two arcs of the circle intercepted by the secant and the tangent have measures in a 7:3 ratio. Find the measure of the third arc

Answer 1

The measure of the third arc is
`150\°`

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

`42\°=(1)/(2)(x-y)`

`84\°=(x-y)` ----> equation A

`(x)/(y)=(7)/(3)`

`x=(7)/(3)y` -----> equation B

Substitute equation B in equation A and solve for y

`84\°=((7)/(3)y-y)`

`84\°=((4)/(3)y)`

`y=3*84\°/4=63\°`

Find the value of x

`x=(7)/(3)(63\°)=147\°`

step 2

Find the measure of the third arc

Let

z------> the measure of the third arc

we know that

`x+y+z=360\°` -----> complete circle

substitute the values and solve for z

`147\°+63\°+z=360\°`

`z=360\°-(147\°+63\°)=150\°`