Question

Find the value of x and then the measure of Arc XYZ and Arc XZ. Show work on a piece of paper.

Answer 1

Given:

The measure of arc XY is (31x)°

The measure of arc YZ is (35x - 16)°

We need to determine the value of x, measure of arc XYZ and arc XZ.

Value of x:

From the figure, it is obvious that the arcs XY and YZ are congruent.

Thus, we have;

$31x=35x-16$

Subtracting both sides by 35x, we have;

$-4x=-16$

   $x=4$

Thus, the value of x is 4.

Measure of arc XYZ:

The measure of arc XYZ is given by

$m \widehat{XYZ}=m \widehat{XY}+m \widehat{YZ}$

The measure of arc XY is given by

$m \widehat{X Y}=31(4)=124^{\circ}$

The measure of arc YZ is given by

$m \widehat{Y Z}=35(4)-16=124^{\circ}$

Hence, the measure of arc XYZ is given by

$m \widehat{XYZ}=124^{\circ}+124^{\circ}=248^{\circ}$

Therefore, the measure of arc XYZ is 248°

Measure of arc XZ:

The measure of arc XZ is given by

$m \widehat{XZ}=360^{\circ}-m \widehat{XYZ}$

Substituting the values, we have;

$m \widehat{XZ}=360^{\circ}-248^{\circ}$

$m \widehat{XZ}=112^{\circ}$

Thus, the measure of arc XZ is 112°