Question

in circle y, what is mtu? 82° 100° 106° 118°

Explanation:

Step1: Recall circle - arc sum property

The sum of the measures of the arcs of a circle is 360°.

Step2: Set up an equation

Let \(m\widehat{TU}=x\). We know that \(m\widehat{RS} = 106^{\circ}\), \(m\widehat{ST}=94^{\circ}\), and \(m\widehat{TU}=x\), and \(m\widehat{UR}\) is part of the circle. Since the sum of the arc - measures around a circle is 360°, we have \(106 + 94+x+m\widehat{UR}=360\). But if we assume the circle is divided into four arcs and we want to find \(x\), and since the circle's arc - measure sum is 360°, we can also use the fact that the opposite arcs of a cyclic quadrilateral - like the one formed by the points on the circle (in a sense related to the arc - measure relationships) or just the general arc - sum property. We know that \(106+94 + x+(360-(106 + 94+x))=360\). In a more straightforward way, \(106+94+x + y=360\) (where \(y\) is the remaining arc). Since we are interested in \(x\), we can calculate as follows: \(x=360-(106 + 94+42)\) (assuming the circle is divided into four arcs and we can find the missing arc by subtracting the known arcs from 360). First, add the known arcs: \(106+94 = 200\). Then, \(360-200 - 42=118\).

Answer:

118°