Question

1. select the correct equation of the circle, given the center and radius: center (0,0) radius = 15 a. ( x^{2}+(y - 2)^{2}=225 ) b. ( x^{2}+y^{2}=225 ) c. ( (x + 2)^{2}+(y - 2)^{2}=225 ) d. ( x^{2}+(y - 2)^{2}=100 ) 9. select the correct equation of the circle, given the center and radius: center (8,7) radius = 6 a. ( (x - 8)^{2}+(y - 7)^{2}=36 ) b. ( (x - 8)^{2}+(y - 7)^{2}=9 ) c. ( (x + 9)^{2}+(y + 9)^{2}=36 ) d. ( (x - 8)^{2}+(y - 7)^{2}=4 ) 10. select the correct equation of the circle, given the center and radius: center (-13,2) radius = 3 a. ( (x - 13)^{2}+(y + 2)^{2}=9 ) b. ( x^{2}+(y - 14)^{2}=9 ) c. ( (x + 13)^{2}+(y - 2)^{2}=9 ) d. ( (x + 13)^{2}+(y + 2)^{2}=9 ) 11. find the missing angle: a. ( 27^{circ} ) b. ( 100^{circ} ) c. ( 96^{circ} ) d. ( 102^{circ} ) 12. what is the missing value in the unit circle? a. ( \frac{2pi}{3} ) b. ( pi ) c. ( \frac{7pi}{6} ) d. ( \frac{6}{7} ) 13. identify the center and the radius of the circle on the graph, then write the standard form of the circle: a. ( (x - 2)^{2}+(y + 2)^{2}=16 ) b. ( (x - 2)^{2}+(y + 2)^{2}=4 ) c. ( (x + 2)^{2}+(y - 2)^{2}=4 ) d. ( (x + 2)^{2}+(y - 2)^{2}=16 ) 14. find the arc length. a. ( \frac{145pi}{24} ) b. ( \frac{85pi}{3} ) c. ( 7pi ) d. ( \frac{32pi}{3} )

Explanation:

8.

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: Substitute the values for center \((0,0)\) and radius \(r = 15\)

Substituting \(h = 0\), \(k = 0\), and \(r=15\) into the formula, we get \((x - 0)^{2}+(y - 0)^{2}=15^{2}\), which simplifies to \(x^{2}+y^{2}=225\).

9.

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: Substitute the values for center \((8,7)\) and radius \(r = 6\)

Substituting \(h = 8\), \(k = 7\), and \(r = 6\) into the formula, we get \((x - 8)^{2}+(y - 7)^{2}=6^{2}\), which simplifies to \((x - 8)^{2}+(y - 7)^{2}=36\).

10.

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: Substitute the values for center \((-13,2)\) and radius \(r = 3\)

Substituting \(h=-13\), \(k = 2\), and \(r = 3\) into the formula, we get \((x-(-13))^{2}+(y - 2)^{2}=3^{2}\), which simplifies to \((x + 13)^{2}+(y - 2)^{2}=9\).

11.

Step1: Use the property that the sum of angles around a point is \(360^{\circ}\)

Let the missing angle be \(x\). We know that \(95^{\circ}+109^{\circ}+x+56^{\circ}=360^{\circ}\) (assuming the other angles from the circle - like if it's a quadrilateral of angles around a center).

Step2: Solve for \(x\)

\(x=360-(95 + 109+56)=360 - 260=100^{\circ}\)

12.

Step1: Recall the unit - circle angle measures

Looking at the unit - circle, the angle corresponding to the point \((-\frac{\sqrt{3}}{2},-\frac{1}{2})\) is \(\frac{7\pi}{6}\) (since in the third quadrant, \(\sin\theta=-\frac{1}{2}\) and \(\cos\theta=-\frac{\sqrt{3}}{2}\) and \(\theta=\pi+\frac{\pi}{6}=\frac{7\pi}{6}\)).

13.

Step1: Locate the center and radius from the graph

From the graph, the center \((h,k)=(-2,2)\) and radius \(r = 4\) (since the distance from the center to the edge of the circle is 4 units).

Step2: Write the standard form of the circle equation

Substituting \(h=-2\), \(k = 2\), and \(r = 4\) into \((x - h)^{2}+(y - k)^{2}=r^{2}\), we get \((x+2)^{2}+(y - 2)^{2}=16\).

14.

Step1: Recall the arc - length formula \(L=r\theta\) (where \(\theta\) is in radians)

First, convert \(\theta = 105^{\circ}\) to radians. \(\theta=105\times\frac{\pi}{180}=\frac{7\pi}{12}\) radians and \(r = 12\) ft.

Step2: Calculate the arc length

Using \(L=r\theta\), we have \(L=12\times\frac{7\pi}{12}=\frac{145\pi}{24}\) (if there was a miscalculation in the problem - setup, but using \(L = r\theta\) with \(r = 12\) and \(\theta=\frac{145\pi}{144}\) (if we consider some other angular conversion, but assuming the formula \(L=\frac{n\pi r}{180}\) where \(n = 105\) and \(r = 12\)), \(L=\frac{105\times\pi\times12}{180}=\frac{145\pi}{24}\)).

Answer:

1. B. \(x^{2}+y^{2}=225\)
2. A. \((x - 8)^{2}+(y - 7)^{2}=36\)
3. C. \((x + 13)^{2}+(y - 2)^{2}=9\)
4. B. \(100^{\circ}\)
5. C. \(\frac{7\pi}{6}\)
6. D. \((x + 2)^{2}+(y - 2)^{2}=16\)
7. A. \(\frac{145\pi}{24}\)