Question

Study the diagram of circle C. In circle C, a circumscribed angle, ∠XYZ, and a central angle, ∠ZCX, intercept XZ⌢. PLS HELP I NEED AN ANSWER ASAP!!!!

Answer 1

Answer:

x = 19

Step-by-step explanation:

According tp the circumscribed angle theorem

A circumscribed angle is supplementary to the central angle that intercepts the same arc

The circumscribed angle is XYZ and the central angle is ZCX

We require to find the measure of arc XZ

The sum of the arcs on a circle is 360° , then

XZ = 360° - m XPZ = 360° - 271° = 89°

A central angle is equal in measure to the arc that subtends it , then

∠ ZCX = m XZ = 89°

∠ ZCX + ∠ XYZ = 180° , that is

89 + 4x + 15 = 180

4x + 104 = 180 ( subtract 104 from both sides )

4x = 76 ( divide both sides by 4 )

x = 19

Answer 2

Answer:

x = 19

Step-by-step explanation:

According to the Intersecting Tangents Theorem, the measure of the angle formed by two tangents that intersect at a point outside a circle is equal to one-half of the positive difference of the measures of the intercepted arcs.

In this case, the circumscribed angle is ∠XYZ and the intercepted arcs are major arc XPZ and minor arc ZX. Therefore:

$m\angle XYZ = \dfrac{1}{2}\left(\overset{\frown}{XPZ}-\overset{\frown}{ZX}\right)$

The sum of the arcs of a circle is always equal to 360°. Therefore, if the measure of major arc XPZ is 271°, then:

$\overset{\frown}{ZX}= 360^{\circ}-\overset{\frown}{XPZ}\\\\\overset{\frown}{ZX}= 360^{\circ}-271^{\circ}\\\\\overset{\frown}{ZX}= 89^{\circ}$

Substitute the values into the equation and solve for ∠XYZ:

$m\angle XYZ = \dfrac{1}{2}\left(271^{\circ}-89^{\circ}\right)\\\\\\m\angle XYZ = \dfrac{1}{2}\left(182^{\circ}\right)\\\\\\\m\angle XYZ =91^{\circ}$

Now, to find the value of x, set the angle expression for angle XYZ equal to 91° and solve for x:

$(4x+15)^{\circ}=91^{\circ}\\\\\\4x+15=91\\\\\\4x=91-15\\\\\\4x=76\\\\\\\dfrac{4x}{4}=\dfrac{76}{4}\\\\\\x=19$

Therefore, the value of x is:

$\Large\boxed{\boxed{x=19}}$