The radius \(\displaystyle r\) of a circle, given its area \(\displaystyle A\), can be found using the formula 

\(\displaystyle A = \pi r^{2}\)

Set \(\displaystyle A= 81 \pi\):

\(\displaystyle \pi r^{2} = 81 \pi\)

Find \(\displaystyle r\) by dividing both sides by \(\displaystyle \pi\) and then taking the square root of both sides:

\(\displaystyle \pi r^{2} \div \pi = 81 \pi \div \pi\)

\(\displaystyle r^{2} = 81\)

\(\displaystyle r = \sqrt{81} = 9\)

The circumference of this circle is found using the formula

\(\displaystyle C = 2 \pi r\):

\(\displaystyle C = 2 \pi (9)\)

\(\displaystyle C= 18 \pi\)

A \(\displaystyle 90 ^{\circ }\) arc of the circle is one fourth of the circle, so the length of the arc is 

\(\displaystyle L = \frac{1}{4}C\)

\(\displaystyle = \frac{1}{4} \cdot 18 \pi\)

\(\displaystyle = \frac{9}{2} \pi\)

If a sector covers \(\displaystyle \frac{4}{5}\) of the area of a given circle, what is the measure of that sector's central angle?

While this problem may seem to not have enough information to solve, we actually have everything we need.

To find the measure of a central angle, we don't need to know the actual area of the sector or the circle. Instead, we just need to know what fraction of the circle we are dealing with. In this case, we are told that the sector represents four fifths of the circle. 

All circles have 360 degrees, so if our sector is four fifths of that, we can find the answer via the following.

\(\displaystyle \measuredangle =\frac{4}{5}(360^{\circ})=288^{\circ}\)

So our answer is:

\(\displaystyle 288^{\circ}\)

The total number of degrees in a circle is \(\displaystyle 360^{\circ }\), so the shaded sector represents \(\displaystyle \frac{144^{\circ }}{360^{\circ }}\) of the circle. In terms of percent, this is 

\(\displaystyle \frac{144^{\circ }}{360^{\circ }} \times 100 \% = \frac{14,400^{\circ }}{360^{\circ }} \%= 40\%\).

The shaded sector is 40% if the circle, so the unshaded sector is \(\displaystyle 100 \% - 40\% = 60\%\) of the circle.

A congruency statement about two triangles implies nothing about the relationship between two sides of the same triangle, so we can eliminate \(\displaystyle \overline{MO}\). Also, the length of a segment with endpoints on two different  triangles depends on their positioning, not their congruence, so we can eliminate \(\displaystyle \overline{NQ}\).

Since \(\displaystyle N\) and \(\displaystyle O\) are in the same positions in the name of the first triangle as \(\displaystyle Q\) and \(\displaystyle R\), \(\displaystyle \overline{NO}\) and \(\displaystyle \overline{QR}\) are corresponding sides of congruent triangles, and 

\(\displaystyle \overline{NO} \cong \overline{QR}\). 

Since the letters of the name of a segment are interchangeable, this statement can be rewritten as

\(\displaystyle \overline{NO} \cong \overline{RQ}\),

making \(\displaystyle \overline{RQ}\) the correct choice.

One yard is equal to three feet, or thirty-six inches. The three given sidelengths are equal to one another, making this an equilateral triangle.

The altitude perpendicular to the hypotenuse of a right triangle divides that triangle into two smaller triangles similar to each other and the large triangle. Therefore, the sides are in proportion. The hypotenuse of the triangle is equal to

\(\displaystyle \sqrt{5^{2}+12^{2}} = \sqrt{25+144} = \sqrt{169} =13\).

Therefore, we can set up a proportion and solve for \(\displaystyle X\):

\(\displaystyle \frac{X}{12} = \frac{12}{13}\)

\(\displaystyle \frac{X}{12}\cdot 12 = \frac{12}{13} \cdot 12\)

\(\displaystyle X = \frac{144}{13} = 11\frac{1}{13}\)

To find the perimeter of a triangle, we will use the following formula:

\(\displaystyle P = a+b+c\)

where a, b, and c are the lengths of the sides of the triangle.

Now, we know the base of the triangle is 16cm. Because it is an equilateral triangle, all sides are equal. Therefore, all sides are 16cm.  So, we get

\(\displaystyle P = 16\text{cm} +16\text{cm} +16\text{cm}\)

\(\displaystyle P = 48\text{cm}\)

The formula to find the perimeter of an equilateral triangle is

\(\displaystyle P = 3a\)

where a is the length of one side of the triangle. Because it is an equilateral triangle, all sides are equal. Therefore, we can use any side in the formula. To find the length of one side, we will solve for a.

Now, we know the perimeter of the equilateral triangle is 36in. So, we will substitute and solve for a. We get

\(\displaystyle 36\text{in} = 3a\)

\(\displaystyle \frac{36\text{in}}{3} = \frac{3a}{3}\)

\(\displaystyle 12\text{in} = a\)

\(\displaystyle a = 12\text{in}\)

Therefore, the length of one side of the equilateral triangle is 12in.

To find the perimeter of a triangle, we will use the following formula:

\(\displaystyle P = a+b+c\)

where a, b, and c are the lengths of the sides of the triangle.

Now, we know one side of the triangle has a length of 17in. Because it is an equilateral triangle, all sides are equal. Therefore, all sides are 17in. So, we can substitute. We get

\(\displaystyle P = 17\text{in} +17\text{in} +17\text{in}\)

\(\displaystyle P = 51\text{in}\)

The formula to find the perimeter of an equilateral triangle is

\(\displaystyle P = 3a\)

where a is the length of one side of the triangle. Because an equilateral triangle has 3 equal sides, we can use any side in the formula. To find the length of one side, we will solve for a.

Now, we know the perimeter of the triangle is 69in. So, we can substitute and solve for a. We get

\(\displaystyle 69\text{in} = 3a\)

\(\displaystyle \frac{69\text{in}}{3} = \frac{3a}{3}\)

\(\displaystyle 23\text{in} = a\)

\(\displaystyle a = 23\text{in}\)

Therefore, the length of one side of the equilateral triangle is 23in.