Question

thursday

1. \\( \hat { a b } = \\)
2. \\( \angle x = \\)
3. \\( ( x + 5 ) ^ { 2 } + ( y - 3 ) ^ { 2 } = 100 \\)

center =
radius =

Explanation:

1) $\overset{\frown}{AB}$

Step1: Recall the definition of arc measure

The measure of an arc formed by a central angle is equal to the measure of the central angle.

Step2: Identify the central angle

Here, the central angle for arc $\overset{\frown}{AB}$ is $80^{\circ}$. So, $\overset{\frown}{AB}=80^{\circ}$

2) $\angle x$

Step1: Use the inscribed - angle and arc relationship

The measure of an inscribed angle is half the measure of its intercepted arc. The intercepted arc of $\angle x$ is $50^{\circ}$.

Step2: Calculate $\angle x$

By the formula $\angle x=\frac{1}{2}\times$ (measure of intercepted arc). So, $\angle x = 25^{\circ}$

3) $(x + 5)^{2}+(y - 3)^{2}=100$

Step1: Recall the standard form of a circle equation

The standard form of a circle equation is $(x - h)^{2}+(y - k)^{2}=r^{2}$, where $(h,k)$ is the center and $r$ is the radius.

Step2: Find the center

For the equation $(x+5)^{2}+(y - 3)^{2}=100$, we have $h=-5,k = 3$. So the center is $(-5,3)$

Step3: Find the radius

Since $r^{2}=100$, then $r=\sqrt{100}=10$

Answer:

1. $80^{\circ}$
2. $25^{\circ}$
3. Center: $(-5,3)$; Radius: $10$