Can anyone help me with this?
1 answer:
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1.
a.
A secant line is a line which intersects the circle at 2 different points.
A tangent line is a line which has only one point in common with the circle.
Check picture 1: The orange line s is a secant line, the blue line t is a tangent line.
b.
An inscribed angle is an angle formed by using 3 points of a circle.
The main property of an inscribed angle is that its measure is half of the measure of the arc it intercepts.
Check picture 2: If
, then the measure of arc KL is 
.
A central angle is an angle whose vertex is the center of the circle, and the 2 endpoints of the rays are points of the circle.
The main property is: the measure of the central angle is equal to the measure of the arc it intercepts.
Check picture 2
2.
To construct the inscribed circle of a triangle, we first draw the 3 interior angle bisectors of the triangle.
They meet at a common point called the incenter, which is the center of the inscribed circle.
We open the compass, from the incenter, so that it touches one of the sides at only one point. We then draw the circle. (picture 3)
To draw the circumscribed circle, we first find the midpoints of each side. We then draw perpendicular segments through these (the midpoints.) They meet at one common point, which is the circumcenter: the center of the circumscribed circle.
We open the compass from the circumcenter to one of the vertices of the triangle. We draw the circle, and see that it circumscribes the triangle.
(picture 4)
3.
Given an equation of a circle:
.
To determine the center and the radius of the equation we must write the above equation in the form :
.
Then, (a, b) is the center, and r is the radius of this circle. We do this process by completing the square.
Note that
becomes a perfect square by adding 1, and
becomes a perfect square by adding 9.
Thus we have:
Thus, the center is (1, -3), and the radius is 2.
4. Not complete
5.
The radius of the pizza is
.
The surface of a circle with radius r is given by the formula
,
and the circumference is given by the formula C=2πr.
Thus, the area of the whole pizza is given by
(square ft).
Each of the 50 slices, has an area of
(square ft)
Notice that the perimeter (the crust) of a slice is made of 2 radii, and the arc-like part.
The arc is 1/50 of the circumference, so it is
.
So the perimeter of one slice is 65.5+65.5+2.62π=131+2.62π
a.
A secant line is a line which intersects the circle at 2 different points.
A tangent line is a line which has only one point in common with the circle.
Check picture 1: The orange line s is a secant line, the blue line t is a tangent line.
b.
An inscribed angle is an angle formed by using 3 points of a circle.
The main property of an inscribed angle is that its measure is half of the measure of the arc it intercepts.
Check picture 2: If
A central angle is an angle whose vertex is the center of the circle, and the 2 endpoints of the rays are points of the circle.
The main property is: the measure of the central angle is equal to the measure of the arc it intercepts.
Check picture 2
2.
To construct the inscribed circle of a triangle, we first draw the 3 interior angle bisectors of the triangle.
They meet at a common point called the incenter, which is the center of the inscribed circle.
We open the compass, from the incenter, so that it touches one of the sides at only one point. We then draw the circle. (picture 3)
To draw the circumscribed circle, we first find the midpoints of each side. We then draw perpendicular segments through these (the midpoints.) They meet at one common point, which is the circumcenter: the center of the circumscribed circle.
We open the compass from the circumcenter to one of the vertices of the triangle. We draw the circle, and see that it circumscribes the triangle.
(picture 4)
3.
Given an equation of a circle:
To determine the center and the radius of the equation we must write the above equation in the form :
Then, (a, b) is the center, and r is the radius of this circle. We do this process by completing the square.
Note that
Thus we have:
Thus, the center is (1, -3), and the radius is 2.
4. Not complete
5.
The radius of the pizza is
The surface of a circle with radius r is given by the formula
and the circumference is given by the formula C=2πr.
Thus, the area of the whole pizza is given by
Each of the 50 slices, has an area of
Notice that the perimeter (the crust) of a slice is made of 2 radii, and the arc-like part.
The arc is 1/50 of the circumference, so it is
So the perimeter of one slice is 65.5+65.5+2.62π=131+2.62π
This is a sideways opening parabola, opening to the right to be more specific, since the leading coefficient is a positive 1. The rule for a focus and a directrix is that they are the same number of units from the vertex (in other words, the vertex is dead center between them), and that the vertex is on the same axis that the focus is. We need to find the vertex then to determine what the focus and the directrix are. We will complete the square on that to find the vertex. Begin by setting it equal to 0, then move the 2 over by addition to get 
. Now we will complete the square on the y terms. Take half the linear term, square it, and add it to both sides. Our linear term is 14. Half of 14 is 7, and 7 squared is 49. So we add 49 to both sides. 
, which of course simplifies to 
. The purpose of this is to find the k coodinate of the vertex which will be revealed when we write the perfect square binomial we created during this process: 
. Moving the 51 back over by subtraction gives us 
. The vertex then is (-51,-7). The formula to find the focus using this vertex is 
. As I stated quite a while ago, the leading coefficient on our parabola was a +1 so our "a" value is 1, and the focus is then found in 
which simplifies to 
. If the vertex is (-51, -7) and the focus is (-50.75, -7), then the distance between them is 1/4, or .25. That means that the directrix is also .25 units from the vertex, but in the other direction. Our directrix is a vertical line, and it will have the equaion x = -51.25. Summing up, your focus is (-50.75, -7) and your directrix is x = -51.25