Question

21.
Using the formula:
Angle Outside = 1/2(arc-arc).
Find the measure of arc XY.
Also show solving steps.

Answer 1

Answer:

185°

Step-by-step explanation:

According to the Intersecting Secant-Tangent Theorem, if a tangent segment and a secant segment are drawn to a circle from an exterior point, the measure of the angle formed by the two segments is equal to one-half the positive difference of the measures of the intercepted arcs.

In this case, the angle formed by the tangent and secant intersecting is ∠XWZ, and the intercepted arcs are XY and XZ. Therefore:

$m\angle XWZ=\dfrac{m\overset{\frown}{XY}-m\overset{\frown}{XZ}}{2}$

From observation of the given diagram:

$m\angle XWZ = 41^{\circ}$

$m\overset{\frown}{XZ}=103^{\circ}$

Substitute these values into the equation:

$41^{\circ}=\dfrac{m\overset{\frown}{XY}-103^{\circ}}{2}$

Multiply both sides of the equation by 2:

$82^{\circ}=m\overset{\frown}{XY}-103^{\circ}$

Add 103° to both sides:

$82^{\circ}+103^{\circ}=m\overset{\frown}{XY}\\\\\\m\overset{\frown}{XY}=185^{\circ}$

Therefore, the measure of arc XY is:

$\Large\boxed{\boxed{m\overset{\frown}{XY}=185^{\circ}}}$