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Tangent two circles

6:03 PM | Label: Math

Tangent of two circles. Before dividing his case, we note some important points about this chapter (tangent circles). Principal importance as follows:
* The line tangent to the circle perpendicular to a line to the center of the circle (a radius of a circle)
* Remember the Pythagorean formula. Because here will be used hold the Pythagorean formula. Pythagorean formula, namely:

$a^2+b^2=c^2$

a, b and c are the sides of a right triangle. with a and b are the sides of square, and c is the hypotenuse.

* Note also that the number of angles in a triangle is 180 degrees.

Now we will discuss about the tangent of two circles. Divided into 2, namely:
* The line outside the common tangent
* Common tangent line in
* The line outside the common tangent

Note the picture above!
There are two circles, one circle with center $O_1$ and 2 with the center circle $O_2$ . The radius of the circle 1 we write $r_1$ and the radius of circle 2 we write $r_2$ . Long lines outside the common tangent of two circles symbolized our $p.g.l$ . The distance of two circle center (a long line connecting the center of circle 1 and circle 2) we are symbolized as $j$
Which is called the length of common tangent line outside?
Tangent length is measured from the point of tangency on the circle 1 to the point of tangency on the circle 2. In the picture can be written as AB.
To find the length of tangent line of external, can be searched by using the formula:

$(p.g.l)^2=j^2-(r_1-r_2)^2$

$p.g.l= \sqrt{j^2-(r_1-r_2)^2}$

Can we exchange the position $r_1$ and $r_2$ ?
Can. That is, we may write $(p.g.l)^2=j^2-(r_2-r_1)^2$ . It's okay to do. As long as they'll be careful with negative signs. Negative numbers when squared, then the result is positive. Likewise, positive numbers that squared the result is positive. Thus, by swapping the positions $r_1$ and $r_2$ Our calculation will be the same.
Where are these formulas?
The formula was obtained from the Pythagorean formula. We note the external tangent. Remember! tangent line is perpendicular to the radius. Note the picture!
If the tangent line is we are so touched one of the shear center of the circle, it will get a right triangle. By applying the Pythagorean theorem on right-angled triangle, with j is the hypotenuse, and sides of a long tangent penyikunya is out and the excess of his fingers.
Try using the Pythagorean formula, then the result is a formula that has been given.
* Common tangent line in

Tangent line of the two circles is a red line on the image. Of course, to find the length we can use the following formula:

$(p.g.d)^2=j^2-(r_1+r_2)^2$

$p.g.d= \sqrt{j^2-(r_1+r_2)^2}$

$p.g.d$ is the length of the tangent line of
The formula is almost the same, just sign in kurungnya now is positive.

How to memorize!

How do I memorize? How do I differentiate outside the tangent formula and the tangent line in? If we like to memorize, just memorize the formula is only on the positive and negative signs.

OUTSIDE - LESS

IN - ADD

Only the sign was different. Can be memorized in this way. If outside, the word "outside" ends with the letter r. then the formula minus. The word "in" ending in the letter m, then the formula is added.
Most of the errors students
Majority student error is to change the minus sign and added rather than on his r. That is, a sign that replaced it has worked on after j. For example, for the formula in, they write $(p.g.d)^2=j^2+(r_1-r_2)^2$ .. while for the outer formula, they write $(p.g.d)^2=j^2-(r_1-r_2)^2$ ... This is wrong. please be careful here.
Hopefully a little help.

Source : asimtot.wordpress.com

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Circle

3:49 AM | Label: Math

In Euklid geometry , a circle is the set of all points on the field within a certain distance, called the radius, from a certain point, called the center. The circle is an example of simple closed curves , dividing the field into the inside and the outside.

Elements of the circle

The elements contained in the circle, is as follows:

n a point inside the circle is the reference to determine the distance to the set point of the building so that the same circle. Lngkiaran element in the form of points, namely:
1. Center point (P)
  is the distance between the center of the circle the price is constant and is called the radius.

Elements that form a circle ruler, namely:
1. The radius (R)
  is a straight line connecting the center with a circle.
2. Bowstring (TB)
  is a straight line inside the circle which cuts the circle at two different points (TB).
3. Arc (B)
  is a curved line either open or closed which coincides with the circle.
4. Circumference of a circle (K)
  is the longest arc on the circle.
5. Diameter (D)
  is the largest bow string whose length is twice the radius. It shares the same diameter circle area.
6. Apotema
  is the shortest line between the bowstring and the center circle.

Elements that form the circle area, namely:
1. Pie (J)
  is an area of a circle bounded by the arc and two radii that are on both ends.
2. Borderline (T)
  is an area of a circle bounded by an arc with a rope bow.
3. Discs (C)
  represents all areas inside the circle. The extent of the radius squared multiplied by pi. The disc is the largest pie.

Equation

A circle has equation

$(X - x_0) ^ 2 + (y - y_0) ^ 2 = R ^ 2 \!$

with $R \!$ is the radius of the circle and $(X_0, y_0) \!$ are the coordinates of the center circle

Parametric Equations

Circles can also be formulated in an equation parameterik, namely

$x = x_0 + R \ cos (t) \!$

$y = y_0 + R \ sin (t) \!$

which if allowed to undergo t will be a circular trajectory in xy space.

Area of circle

Area of circle having the formula

$A = \ pi R ^ 2 \!$

that can be derived by integrating the area of a circle element

$dA = rd \ theta \ dr$

in polar coordinates, namely

$\ Int dA = \ int_ {r = 0} ^ R \ int_ {\ theta = 0} ^ {2 \ pi} rd \ theta \ dr = \ int_ {r = 0} ^ R RDR \ int_ {\ theta = 0 } ^ {2 \ pi} d \ theta = \ frac 1 2 (R ^ 2-0 ^ 2) \ (2 \ pi-0) = \ pi R ^ 2 \!$

In the same way can also be calculated vast semi-circle, quarter circle, and parts of the circle. Also do not miss to calculate the area of a ring circle with a radius of $R_1 \!$ and outer radius $R_2 \!$ .

Addition section element

Area of circle can be calculated with the cut it up as the elements of a pie for later rearranged into a rectangle whose width can be easily calculated. In pictures r means the same as R is the radius of the circle.

Area of pie

Broad segment of a circle can be calculated if the area of a circle made a function of R and θ, namely;

$A (R, \ theta) = \ frac 1 2 R ^ 2 \ theta \!$

with the restriction θ value is between 0 and 3π. When the value θ 2π, the calculated segment is the largest segment, or area of a circle.

Area of a circle ring

A circular ring has an area that depends on the radius in $R_1 \!$ and outer radius $R_2 \!$ , Ie

$A_ {ring} = \ pi (R_2 ^ 2 - R_1 ^ 2) \!$

where for $R_1 = 0 \!$ this formula back into the formula area of a circle.

Area of a circle cut ring

By combining the two previous formulas, can be obtained

$A_ {cut \ ring} = \ frac \ pi 2 (R_2 ^ 2 - R_1 ^ 2) \ theta \!$

which is the area of a ring was intact.

Around the circle

Circumference of a circle has the formula:

$L = 2 \ pi R \!$

The length of arc

Length of arc of a circle can be calculated using the formula

$L = R \ theta \!$

derived from the formula for calculating the length of a curve

$dL = \ int \ sqrt {1 + \ left (\ frac {dy} {dx} \ right) ^ 2} dx \!$

where used

$y = \ pm \ sqrt {R ^ 2 - x ^ 2} \!$

as a curve in a circle. Sign $\ Pm$ suggests that there are two curves, namely the top and bottom. Both are identical (remember the definition of a circle), so that really only needs to be calculated once and the result multiplied by two.

Source : Wikipedia.org

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Can music Slowing Aging Effects

3:02 AM | Label: Tips and Tricks

Musical instrument played down the effects of aging especially related to hearing. A study in the United States reveals, those familiar with musical instruments throughout his life has the capability of hearing and a better memory than those who never touch the music.

"Practice music may prolong life, improve memory, and improve the ability to hear speech in noise," says Nina Kraus, authors of the study which served as the Director of the Auditory Neuroscience Laboratory at Northwestern University, United States, as quoted by the Daily Mail.

In their study, researchers involved 18 people musicians and 19 non-musicians ages 45 to 65 years. They asked respondents to listen to speech in noisy atmosphere. They also conducted a series of tests such as tests of visual memory, voice recording capability, and ability to hear.

The musicians learn to play musical instruments at an early age or under nine years old and still active until now, managed to beat the non-musicians in all tests. However, in tests of visual memory ability, both groups showed almost identical ability.

Doctor Kraus said, the experience of extracting sound from a wide range of musical instruments and sound sequences given the increasing development of auditory skills. "Playing music involves their ability to retrieve relevant patterns, including their own voices, instruments, harmony, and rhythm."

Dr. Kraus said, practicing music can improve memory and auditory nervous system. "The experience of musical communication can combat the problems associated with age," he said, adding that the study was published in the scientific journal PLoS One recent issue.

Strengthening these findings, other studies revealed that giving children music lessons could become a training to increase the concentration of learning in the school. Learning music can also reduce some negative effects of aging.

Source : vivanews.com

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