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

Answer:

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

$42\°=\frac{1}{2}(x-y)$

$84\°=(x-y)$ ----> equation A

$\frac{x}{y}=\frac{7}{3}$

$x=\frac{7}{3}y$ -----> equation B

Substitute equation B in equation A and solve for y

$84\°=(\frac{7}{3}y-y)$

$84\°=(\frac{4}{3}y)$

$y=3*84\°/4=63\°$

Find the value of x

$x=\frac{7}{3}(63\°)=147\°$

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

$147\°+63\°+z=360\°$

$z=360\°-(147\°+63\°)=150\°$