The problem of chord centrity, what is the chord centroid formula?

Make AE perpendicular to BC to BC to E

Because ab=5, ac=12, and the triangle abc is a right triangle.

So bc=13 (Pythagorean theorem).

Let ae=xthen ce=13-x

5 -x = 12 -(13-x) (2 means squared) is solved to x=25/13

Because ae is the radius. According to the vertical diameter theorem, be=ed

So be = 2 * 25/13 = 50/13, so cd = 13-50/13 = 119/13

From the Pythagorean theorem: bc=13

The intersection BC with AB as the radius is d=>ad=5

In triangular ACD, cosc=12 13, ac=12, ad=5> using cosine theorem: cosc=ac2+cd 2-ad 2 2*ac*ad, is a bit annoying, but it is much simpler than using chord centerdistance.

The chord centroid distance formula: d=|ax0+by0+c|/√a^2+b^2)。

The length of the perpendicular segment from the center of the circle to the chord is called the chord centroid distance of this string. Equality relationship between the central angles, arcs, chords, and chord centrax: In the same circle or equal circle, the arc, chord, and chord centroid of the opposite chord are equal to the arc, string, and chord centroid distance of the opposite string, and if one of the four is equal, then the other three are equal.

In the same circle or equal circle, for two unequal strings, their chord centroid distance is also unequal, and the chord centricity of the large string is inversely smaller.

Diameter, chord, nature of arc

1. In a circle, if the diameter of the string is perpendicular to the string, then the diameter bisects the chord and the chord to which the string is opposed.

2. In a circle, if the diameter of the chord is bisected (the string itself is not the diameter), then the diameter is perpendicular to the chord, and the arc to which the string is directed is bisected.

3. In a circle, if the diameter bisects the arc, then the diameter bisects the chord against which the arc is opposite.

4. Within the circle, the perpendicular bisector of the string passes through the center of the circle.

The chord centroid formula is: OC= R2-AC2.

The coordinates of the intersection of the line and the circle a, b, and ab are c, and oc is perpendicular to ab.

Chordal centroid distance: OC= R2-AC2

The easy way to do this is to use the formula d=| for the distance from p(x0,y0) to the straight line ax+by+c=0ax0+by0+c|/√a^2+b^2)。

Note that the condition is a, b≠0, equal to 0, do not use this formula.

Properties of the central angle, arc, chord, chord centricity:

In the same circle or equal circle, if the angles of the center of the circle are equal, then the arcs to which they are opposite, the chords to which they are opposite, and the centricities of the strings to which they are opposed are equal (the inverse proposition also holds).

If the central angles of the circles are unequal, then the opposite arc with a large central angle is larger, the opposite chord is larger, and the chord center distance on the opposite string is small (the inverse proposition is also true).

The chord centroid formula is: OC= R2-AC2.

The length of the perpendicular segment from the center of the circle to the chord is called the chord centroid distance of this string. Rounded corners.

The equality relationship between the arc, the chord, and the chord centroid are equal, in the same circle or equal circle, the arc, chord, and chord centroid distance of the opposite string are equal, and if one of the four is equal, then the other three are equal. In a circle, the distance from the center of the circle to any string of the circle is called the chord centroid distance of that string.

Circle

Within a plane, around a point and with a certain length.

A closed curve formed by a degree of rotation is called a circle. In a plane, a circle is a set of points whose distance from a fixed point is equal to a fixed length, and a circle has an infinite number of axes of symmetry.

The axis of symmetry has rotational invariance through the center of the circle, and the circle is a conic curve.

It is obtained from a planar truncated cone parallel to the bottom surface of the cone.

The circle can be seen as being infinitely smaller.

points of the regular polygon.

When the polygon has more sides, its shape, circumference, and area are closer to a circle. So, there is no real circle in the world, and the circle is really just a conceptual figure. (When a straight line becomes a curve, it is an infinite point, so it can also be said to be an absolute circle).

The above content reference:Encyclopedia – Circle

Chord centroid formula: OC= R2-AC2.

The center distance of the chord is the distance from the midpoint of the chord to the center of the circle (using the formula for the distance between two points), and it is also equal to the distance from the center of the circle to the line where the string is located (using the formula for the distance from the point to the line).

The coordinates of the intersection of the line and the circle a, the midpoint is c, oc is perpendicular to ab, and the chordal centroid oc = r 2-ac 2.

The easy way to do this is to use the formula d=| for the distance from p(x0,y0) to the straight line ax+by+c=0ax0+by0+c|/√(a^2+b^2)

Note that the condition is a, b≠0, equal to 0, do not use this formula.

Related formula calculations.

Radius of the circle: r

Diameter: dPi: (The value is between to.......)infinite non-cyclic decimals), usually taken as a numeric value of .

Circle area: s= r ; s=π(d/2)²

Area of the semicircle: s semicircle = ( r ; )/2

The area of the ring: s big circle - s small circle = (r -r) (r is the radius of the great circle, r is the radius of the small circle).

The circumference of the circle: c = 2 r or c = d

The circumference of the semicircle: d+(d)2 or d+r

1. Definition and explanation of knowledge points.

The chord centroid formula is a mathematical formula that describes the relationship between the chord length of a circle and the angle of the center of the circle to which the string corresponds. The chord centroid distance refers to the vertical distance from the center of the circle to the chord. In geometry, this formula can be used to calculate the chord length for a given radius and central angle, or the central angle for a given chord length and radius.

2.Application of knowledge points.

The application of the chord centroid formula includes the following aspects:

1.Calculate chord length: Knowing the radius and central angle of the circle, the corresponding chord length can be calculated by the chord centroid formula.

2.Calculate the central angle: Knowing the radius and chord length of the circle, the corresponding central angle can be calculated by the chord centerdistance formula.

3. Explanation of knowledge points and examples.

Example: A circle with a radius of 10 cm is known, and the length of one of the strings is 12 cm. Find the central angle of the string.

Answer: According to the information given in the question, the radius of the circle is 10 cm and the length of the string is 12 cm. We need to solve for the central angle of the string.

According to the formula of chord centroid distance, there is a relationship between the chord length 2r*sin(2) and the central angle of the circle.

Substitute the known data into the formula: 12 = 2 * 10 * sin(2).

Simplification: sin( 2) =

Using the inverse function sin (-1), we can solve for the value of 2.

sin^(-1)(

Because the solution is 2, the final central angle is = 2 * = .

Therefore, the central angle of the string is about.

The chord centroid distance refers to the distance from the center of the circle to the chord. The chord centroid formula is d=|ax0+by0+c|(a +b), where (x0,y0) is the coordinates of the intersection of the line and the line, a, b are the coefficients in the equation y=kx+b, and c is the constant term in the equation ax+by+c=0. This formula can be used to solve for the chord centroid distance between any two lines in a circle.

Application of knowledge points.

In practical problems, the chord centering formula can be used to solve the chord centering between any two straight lines in a circle, such as solving the express delivery problem in the circular area, the mobile phone signal coverage problem in the circular area, etc. By solving for chord distance, you can better understand the distance between two points in a circular area, so you can optimize the distribution route or signal coverage.

Explanation of knowledge points and examples.

Example: Knowing that the coordinates of the center of the circle are (0,0), the radius is 5, and the equation for the line is y=x, find the chord centroid distance between the line and the circle.

Solution: First, the linear equation y=x is substituted into the chord centroid formula, and the chord centroid formula is d=|ax0+by0+c|/√(a²+b²)。Because the straight line crosses the origin, c=0.

Thus, the chord centroid formula becomes d=|ax0+by0|/√(a²+b²)。

Next, substituting the coordinates of the center of the circle (0,0) and the linear equation y=x gives d=|ax0+by0|/√(a²+b²)=|0×0+0×0|/√(1²+1²)=0。

Therefore, the chordal centroid distance between the line and the circle is 0.

Chord centroid formula: OC= R2-AC2.

The center distance of the chord is the distance from the midpoint of the chord to the center of the circle (using the formula for the distance between two points, it is also equal to the distance from the center of the circle to the line where the string is located (using the formula of the distance from the point to the line).

The coordinates of the intersection of the line and the ruler are a, the midpoint is c, oc is perpendicular to ab, and the chordal centroid oc = r 2-ac 2.

The easy way to do this is to use the formula d=| for the distance from p(x0,y0) to the straight line ax+by+c=0ax0+by0+c|Sold in the world (A 2 + B 2).

Note that the condition is a, b≠0, equal to 0, do not use this formula.

Related formula calculations.

Radius of the circle: r

Diameter: d pi.

Values are ...... toinfinite non-cyclic decimals), usually taken as a numeric value of .

Circle area. s=πr²；s=π(d/2)²

Area of the semicircle: s semicircle = ( r ; )2

The area of the ring: s big circle - s small circle = (r -r) (r is the radius of the great circle, r is the radius of the small circle).

The circumference of the circle: c = 2 r or c = d

The circumference of the semicircle: d+(d)2 or d+r

Chinese name: chord center distance formula.

Meaning: The distance from the center of the circle to the string.

Concept: Circle center, arc, chord.

Chordal centroid distance: OC= R2-AC2

For p(x0,y0), it is the distance to the straight line ax+by+c=0 with the formula d=|ax0+by0+c|a 2+b 2) The distance from the center of the circle to the chord is called the chord centr.