Question

Find the value of x.

Answer 1

74 degrees

Explanation:

The arc degree measurement of the arc that is colored in orange is given as 254 degrees.

The remaining part is the part in green.

The green part plus the orange part should equal 360 degrees because that would make a full rotation around the circle.

`360-254=106`.

The green arc has measurement 106 degrees.

We can find x by computing half the difference of the arcs there.

That is,

`x=(1)/(2)(254-106)`

`x=(1)/(2)(148)`

`x=74`

Answer 2

74°

Explanation:

There are at least a couple of ways to get there.

1) Find the central angle between the radii to the points of tangency. It will be ...

360° - 254° = 106°

The measure of x is the supplement of this:

x = 180° -106° = 74°

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2) Find half the difference of the subtended arcs. The arc shown is 254°; the one not shown is 360° -254° = 106°. Half the difference of these is ...

(254°-106°)/2 = 148°/2 = 74°

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3) Perhaps more simply, we can find this measure symbolically. Let y represent the large arc shown as 254°. Then the smaller one is 360°-y and their difference is ...

difference = y - (360° -y) = 2y -360°

Half this value, the value of angle x is ...

(2y -360°)/2 = y -180°

For y=254°, this is ...

x = 254° -180° = 74°