Connect The Dots: Lines
A line is a connected set of points between two extremes. The points could be in any location. If they all are in one location then that is one point or a line that has zero length. To deal properly with lines, you have to use a kind of location finding algorithm. We can use the rectangular Cartesian coordinate system. In this system, 2 or 3 number designate how far a point is from a chosen reference point that stands as a beginning point. The numbers are put in brackets and separated by commas like this: (x,y,z). The letters x, y and z represent three coordinates on three axis , which are lines mutually perpendicular to each other and such that they meet at on point called the 'origin'.
A line could be straight or zig-zag or a curve. There could be an equation behind it, such as one relating x and y only:
y = x2
To draw this line, you could select certain points in x and find the y-points. You could say x =[0,1,2,3,4,5,6,7,8] for example and for each point y=[0,1,4,9,16,25,36,49,64]
For lessons in graphs and vectors look at: |General Graphing Vectors
Straight Lines as Tangents
One More Dot To Be Sure: Triangles
A triangle is a plane shape formed by three points where each pair of points is connected with a straight line. A triangle can be seen as a shape made by two equally tall rectangles cut in half along their diagonals and joined along their heights. The area of a triangle is:
A = (1/2) (base-lenght)(vertical-height)
Burn A Hole:Circles
If a line fixes one of its end points and uses it and swings itself in one direction on a flat table so that it doesn't turn back until it hits its original position it has drawn an imaginary circle with the other end. A circle is a line drawn using a constant lenght from a chosen fixed point. That is, a circle is the locus of all points drawn from a chosen point using a constant distance from that point. This distance is called a radius. You can also say it is a line drawn by the end-points of a line that turns half-way on a plane using its mid-point.
The fixed distance is called a 'radius' and the 'diameter' is two times the radius.
PI
The ratio of the circumference of a circle, which is the perimeter of the circle, to the diameter is a number called PI. The value of PI was first calculated by Archimedes. The number of digits after the decimal point that make up the number has been grown in size to as much as 20000 digits.
You can use the diagram to find the circumference of a circle.
Each Triangle has a lenght L = 2rsin(dx/2)= 2r(dx)/2
To find the accumulation of all base lenghts, you add these little triangles:
L = 2r(dx)/2 + 2r(dx)/2 + 2r(dx)/2 + 2r(dx)/2...+ 2r(dx)/2 = 2rn(dx)/2
n(dx) = 2(pi)
L = 2r(2(pi))/2 = 2(pi)r
To find the area of a circle we may apply the same diagram:
Each Triangle has a lenght A = r2sin(dx)/2= r2(dx)/2
To find the accumulation of all area, you add these little triangles:
L = r2(dx)/2 + r2(dx)/2 +.. r2(dx)/2= r2n(dx)/2
n(dx) = 2(pi)
L = r22(pi)/2 = (pi)r2
There are ways to do this with integral calculus.
Circles Are Triangles and Quads
You can put a circle in a triangle or a triangle in a circle. In the first case you are circumscribing the triangle while in the second case you are inscribing the the circle. The method to circumscribe a triangle would be to find the mid-points of two sides and take perpendicular bisections which will intersect somewhere on the and this point serves as the anchor of a compass. When you touch your pencil point on a vertex of the triangle and you drwa out a circle you will end up with a circle that touches all vertices of the triangle.
In order to inscribe a circle in a triangle, you have to bisect the angles of two vertices and use the intersections of the two line as an anchor for your compass.
Triangle Ratios
When you put the radius of a circle in the first quadrant, you will have circular ratios of angles between 0 and 90 degrees.
sines
A sine of an angle is how well vertically up or down the radius is inclined. Generally, when the radii line isn't treated like a radius, it just means how well up that line is inclined. Look at the diagram below:
You will see that AD is completely vertical. BC creates more vertical displacement than all the other non-vertical lines. The height AD is created by the combination of the verticals of BC and AB. The angles made by BC is more than that of AB and EB and thus it creates a larger vertical rise.
cosines
The cosine of an angle is well forward or backward the radius is inclined. Looking at the diagram again.
You will see that AD produces no horizontal forward or backward displacements. BC produces less of a horizontal displacement than AB or EB. DC is a sum of the horizontal displacements of EB and BC.
Tangents
The tangent of an angle is how well up or down a chosen forward displacement of a constant radius will produce. It is the ratio of a sine and a cosine.
The Sum of Angles
The sum of angles in a triangle is equal to 180 degrees. The diagram below should explain it.
Its check time! Ask your favorite math teacher to explain the diagram to you. Or wait a few weeks.
The right-angled Triangle
A triangle that has one angle equal to 90 degrees. If you put one of its flat sides horizontal you could conveniently use the triangle ratios to complete its sides and angles if you had just some parts of it.
Finding Sides and angles
For a right-angled triangle, you just have to apply the ratios directly because already the two flat sides are acounted for.
cosines rule
The cosines rule for a triangle requires you have two known sides and the angle between them.
The formula for solving triangles with the cosine rule is:
a2 = b2 + c2 - b2c2cosA
This is for when side 'a' is sought after. If it were 'b' or 'c' we needed, you would just interchange the letters and replace cosA with cosB or cosC.
The cosine rule for finding angles can be used for cases where all the sides are known:
cosA = (b2 + c2 - a2)/( b2c2)
Here too, you have to interchange the letters for different angles.
sines rule
Sine rule goes like this:
sinA/a = sinB/b = sinC/c
You apply the sine rule when you have two sides and one angle or two angles and one side.
Circles in Triangle
You can place a circle in a triangle. How? By bisecting any two angles and using the intersection of those two bisections as the center of the circle.
Triangles in Circles
To put a triangle in a circle, you bisect the sides of the triangle and use the intersections of two bisectors as the center of the circumscribing circle.
Quads in Circles
Now if you put a quadrilateral in a circle certain things come into play. Some rules to such a problem concerning angles which I will prove later are written below:
The cyclic quad we are considering here is ABCD. But there are also ACED and ACOD. Any quad that has its corners touching the circumference of a circle is a
is a cyclic quad.
The sum of all angles in a cyclic quad is 3600
A + B + C + D = 3600
The angles on opposite corners in a cyclic quad are summing up to 1800
A + C = 1800
B + D = 1800
An angle substended to the circumference by the angle at center is half of the angle at the center
On = 2V
O = 2E
X is the exterior angle that is supplementary to D. That is, X + D = 1800
Any exterior angle is equal to the interior angle on the opposit side
X = A
Here are two questions:
CE is a diameter. find the sum of N and P?
Find the sum of H and L?
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