I have 2 questions. PLEASE HELP!?

Question

1-
Given a circle with an inscribed equilateral triangle with each side of the triangle having a measurement of 18 cm. What is the probability of selecting a point at random inside the triangle, assuming that the point cannot lie outside the circular region. Leave your answer in exact form. To receive full credit, please show your work.

2-
In a circle, the radius is 68 units and a chord of the circle is 120 units.
Part A: How far is the chord from the center?
Part B: Find the measure of the arc created by the chord.
To receive full credit, please show your work.

Answer 1

Use the law of sines to find the radius of the circle.
sin 120 / 18 = sin 30 / r
r = 18*sin 30 / sin 120 = 9 / (1/sqrt(2)) = 9sqrt(2)
Now find the area of both the triangle and the circle.
area of circle: pi*(9*sqrt(2))^2 = 162*pi
First find the height of the triangle: h = sqrt(18^2 - 9^2) =9sqrt(3)
Area of the triangle: (1/2)*18 * 9 sqrt(3) = 81sqrt(3)
So the probability of selecting the point inside the triangle is:
81sqrt(3)/(162*pi) = sqrt(3)/(2pi) = .27566

2.
A. the chord is sqrt(68^2 - 60^2) = 32 units from the center.
B. sin(1/2*theta) = 60/68
theta = 123.855 degrees
arc length: 2*pi*68*123.855/360 = 146.994 units.

Answer 2

Both problems use the fact that a segment from the center of a circle to the midpoint of a chord is perpendicular to the chord. Thus, this segment, half the chord, and a radius from the center to the other endpoint of the half-chord form a right triangle.

1. Sketch the circle with the inscribed triangle. Draw the segment from the center of the circle to the midpoint of one side of the triangle (a chord) and the radius from the center to one end point of the side. The probability you want will be p = (area of triangle)/(area of circle). The right triangle described above is a 30-60-90 triangle with the root 3 side = 9. The shortest side is 3sqrt(3) and the hypotenuse is 6sqrt(3). From this, you can find both areas and simplify.

2. This is actually a little easier. Draw the segment from the center of the circle to the midpoint of the chord. It is the length of this segment that you want. The radius from the center of the circle to one endpoint of the chord is 68cm. The half-chord is 60 cm. use the Pythagorean theorem to complete the problem.

Answer 3

The altitude of the equilateral triangle is 9 * sqrt(3) so its area is 9*9*sqrt(3)/2 = 40.5 *sqrt(3).
The radius of the circle = 2/3 * 9*sqrt(3) = 6*sqrt(3)
So area of circle is 36*3 *pi = 108pi
The area inside the circle and outside the triangle is
108pi-40.5sqrt(3)
Probability point will lie in triangle is
[ 40.5*sqrt(3)]/(108*pi).

The Draw a line _|_ to chord. Draw radii from center of circele to chord endpoints. A right triangle is formed with hypotenuse = radius = 68 and long leg = 1/2 chord = 60.
Thus distance from chord to center is sqrt(68^2-60^2) = 32.

60/68 = sin x so x = arcsin(15/17)
arc = 2x = 2arcsin x(15/17) = approx =123.86 degrees