Geometry II
Circles
A circle is the set of all points that are the same distance from a common center.
A line connecting any two points on the circle is called a chord.
The radius is the distance from the center of the circle to any point on the circle.
The distance from one point of the circle through the center to another point is called the diameter. It is equal to twice the radius,
A line that intersects the circle at exactly one point is a tangent.
Example:
radius = 6
diameter = 12
AB is a chord
Point A is a tangential point to the circle
An angle that has its vertex in the center is called a central angle. The central angle describes two arcs. The smaller arc is called the minor angle and larger angle is called the major angle. The central angle and minor arc describe a sector of the circle.
An angles that has its vertex on the circumference of the circle is called an inscribed angle.
The circumference is the length of the outer edge.It is equal to 2 Π.
C= 2Π
The area of the circle is:
A = Πr²
If we want to find the length of an arc, we can set up a ratio.
| central angle | length of arc | |
| 360 | = | circumference |
If we want to find the area of a segment we can set up a ratio.
| central angle | area of sector | |
| 360 | = | area of circle |
Example:
If the area of a circle is equal 36Π what is length of minor arc X ?
Solution:
Since,
A = Πr²
Πr² = 36Π
r = 6
Now we use this ratio:
| central angle | length of arc | |
| 360 | = | circumference |
| 60 | length of arc | |
| 360 | = | 2Π(6) |
length of arc = 1/6 x 12Π = 2Π
If an inscribed angle lies on the same arc as a central angle it is exactly half the central angle.
Here the central angle is 60, the inscribed angles that lie on the same shared arc are both half of 60 or 30.
An inscribed triangle that lies on the diameter of the circle must be a right triangle.
Example:
If PQ || OR and the diameter OR is equal to 18. What is the length of PQ?
a)2Π
b)9Π/4
c)7Π/2
d)9Π/4
e)3Π
Solution:
Since we have two parallel lines angle P must be 35 as well by the Z rule. Now we have two inscribed angles of 35 degrees. The central angle of each must be 70. There are a total of 180 degrees in a half circle. So to find the angle in between we subtract the 2 inscribed angles from 180.
180 - 70 - 70 = 40
Now to find the length of the arc PQ we use the ratio we set up earlier.
| central angle | length of arc | |
| 360 | = | circumference |
| 40 | length of arc | |
| 360 | = | 2Π(9) |
length of arc = 1/9 x 18Π
length of arc = 2Π
The correct answer is A.
Solids
Rectangular Solid
The volume of a rectangular cube is w x l x h.
The above solid’s area is:
w x l x h = 4 x 3 x 6 = 72
The surface area is equal to 2(wl + wh + lh)
It’s just the sum of the 2 dimension outer surfaces.
2(12 + 24 + 18) = 108
The diagonal of a rectangular solid goes from one corner to the opposite corner diagonally. It is the longest distance inside a rectangular solid.
To find the diagonal of a rectangular solid we have:
diagonal = √(w² x l² x h²)
For example, √(4² x 3² x 6²)
Cylinder
The volume of the cylinder is the area of the circular base times the height.
V = Πr² x H
=36Π
The surface area is the area of the top and bottom bases time the circumference time the height.
S = 2Πr² x 2Πr X H
S = 18Π x6Π x 4
Coordinate Geometry
A coordinate plane lets you plot out points in two dimensions. The horizontal line is call the x axis. The vertical line is called the y axis, We can divide the plane into four quadrants. We write a point on the plane as an ordered x and y pair (x,y).
A line is defined by the formula:
y=mx +b
where b is the y intercept(the point where the line crosses the y axis).M is the slope of the line. The slopes tells us whether a line is diagonal,horizontal, or vertical.
For example, in the graph above the y intercept is 2(the point where it crosses the y axis). The slope of a line can be found if you know any two points on the line.
| slope | = | change in y |
| change in x |
Two points on the line above are (0,2) and (2,0)
slope = (2 - 0)/ (0-2) = -1
The equation of this line can be written as
y = -x + 2
The distance between any two points can be found with the following formula:
distance = √[(y2 - y1)² + (x2 - x1)²]
For example, the distance between (0,2) and (2,0) is
distance = √[(0 - 2)² + (2 - 0)²]
√[(- 2)² + (2 )²]
√[8]
2(√2)
(46)
(28)
(2)