We are given the radius of the circle and the perpendicular distance from its center to the chord, which is all we need to calculate the length of the chord. Using the formula for chord length that involves these two quantities, we find the solution as follows, where is the chord length, is the perpendicular distance from the center of the circle to the chord, and is the radius:

We are given the radius of the circle and the perpendicular distance from its center to the chord, which is all we need to calculate the length of the chord. Using the formula for chord length that involves these two quantities, we find the solution as follows, where is the chord length, is the perpendicular distance from the center of the circle to the chord, and is the radius:

We are looking for ways to add three primes to yield a sum of 43. Two or all three (since an equilateral triangle is considered isosceles) must be equal (although, since 43 is not a multiple of three, only two can be equal).

We will set the shared sidelength of the congruent sides to each prime number in turn up to 19:

By the Triangle Inequality, the sum of the lengths of the shortest two sides must exceed that of the greatest. We can therefore eliminate the first three. , , and include numbers that are not prime (21, 15, 9). This leaves us with only one possibility:

- greatest length 19

19 is the correct choice.

Using the formula for arc length, we can plug in the given angle and radius to calculate the length of the arc that subtends the central angle of the sector. The angle, however, must be in radians, so we make sure to convert degrees accordingly by multiplying the given angle by :

This figure can be seen as a composite of two simple shapes: the rectangle with vertices , and the triangle with vertices .

The rectangle has length and height , so its area is the product of these dimensions, or .

The triangle has as its base the length of the horizontal segment connecting and , which is ; its height is the vertical distance from the other vertex to this segment, which is . The area of this triangle is half the product of the base and the height, which is .

Add the areas of the rectangle and the triangle to get the total area:

One mile is 5,280 feet. The perimeter of the pentagon is one-fifth of this, or feet. Since each side of a regular pentagon is congruent, the length of one side is one fifth-of this, or feet.

The three sides of a scalene triangle have different measures. One measure cannot have is 12, but this is not a choice.

It cannot be true that . Since the perimeter is

, we can find out what other value can be eliminated as follows:

Therefore, if , then , and the triangle is not scalene. 9 is the correct choice.

The vertical segment connecting and can be seen as the base of this triangle; this base has length . The height is the perpendicular (horizontal) distance from to this segment, which is 6, the same as the -coordinate of this point. The area of the triangle is therefore

.

The figure referenced is below.

The arc is of the circle, so the circumference of the circle is

.

The radius is this circumference divided by , or

.

is, consequently, the hypotenuse of an isosceles right triangle with leg length ; by the 45-45-90 Triangle Theorem, its length is times this, or

The radius of a circle with area can be found as follows:

The circle, the central angle, and the chord are shown below:

By way of the Isosceles Triangle Theorem, can be proved equilateral, so , the correct response.