The vertex coordinates of the triangle are a (2, 2), B (4, 2), C (6, 4). Try to reduce △ ABC so that the corresponding side ratio of △ def and △ ABC is 1:2

The vertex coordinates of the triangle are a (2, 2), B (4, 2), C (6, 4). Try to reduce △ ABC so that the corresponding side ratio of △ def and △ ABC is 1:2

The center of the answer is uncertain

Given the vertex coordinates of triangle a (4.2) B (2.2) C (2.6), reduce △ ABC so that the corresponding edge ratio of △ def and △ ABC is 1:2, and write the corresponding vertex coordinates

The origin coordinate is (0,0)
The connecting point a of the origin is taken as the point D
Point B is connected to the origin, and the midpoint is taken as point E
The central point of the connecting point C is the point F
Connect def
The question is simple, but it is difficult to answer
d（2,1）e（1,1）c（1,3）

As shown in the figure, the coordinates of the three vertices of △ ABC are known as a (- 4,0), B (1,0), and C (- 2,6) (1) Find the parabola analytic formula passing through a, B and C; (2) Let the line BC intersect Y-axis at point E, connect AE, and verify that AE = CE; (3) Let parabola and y-axis intersect point D, connect AD and intersect BC at point F. is triangle with vertex a, B and f similar to △ ABC?

And ∵ from the parabola passing through C (- 2,6), ᙽ 6 = a (- 2 + 4) (- 2-1), the solution is: a = - 1
The analytical formula of parabola passing through a, B and C is y = - (x + 4) (x-1), that is, y = - x2-3x + 4
(2) It is proved that: let the function analytic formula of line BC be y = KX + B,
From the meaning of the question, we can get the following conclusion
The analytic formula of straight line BC is y = - 2x + 2
The coordinates of point e are (0,2)
∴.
∴AE=CE.
(3) Similar. The reasons are as follows:
Let the analytic formula of the straight line ad be y = K1X + B1, then, the solution is
The analytical formula of straight line ad is y = x + 4
The analytic expression of a straight line is obtained
The coordinate of point F is ()
Then
And ∵ AB = 5,
∴.∴.
And ∵ ABF = ∵ CBA,  Abf ∵ CBA
The triangle with a, B and F as the vertices is similar to △ ABC
[test points] synthesis problems of quadratic function, undetermined coefficient method, relationship between coordinates of points on curve and equation, Pythagorean theorem, determination of similar triangles
[analysis] (1) the analytical formula of parabola can be obtained by using undetermined coefficient method
(2) The function analytic formula of straight line BC is obtained, then the coordinates of point e are obtained, and then the length of AE and CE is calculated respectively to prove the conclusion
(3) The function analytic formula of ad is obtained, and then the coordinates of point F can be obtained by combining the analytic formula of straight line BC, BF and BC can be obtained respectively according to Pythagorean theorem; the judgment can be made if the meaning of the topic is ∠ ABF = ∠ CBA

The vertex coordinates of the triangle are a (2, 2), B (4, 2), C (6, 4). Try to reduce △ ABC so that the corresponding side ratio of △ def and △ ABC is 1:2

The center of the answer is uncertain

We know that the three vertices a (0,1), B (1,0), C (3 / 2,0) of the triangle ABC pass through the origin of the line L, and let the triangle ab The area of triangle ABC is divided into two equal parts, and the slope of the line L is calculated

The equation of line L is: y = KX, the area of s triangle ABC = the area of s triangle OAC - the area of s triangle OAB = 1 / 2 * (3 / 2-1) = 1 / 4. If the line L intersects AB with E and AC with F, then the area of quadrilateral befc is 1 / 2 * the area of triangle ABC = 1 / 8

When △ ABC is rotated 180 ° around the origin of the coordinates, the change characteristics of the coordinates of each vertex are as follows______ .

∵ △ ABC rotates 180 ° around the coordinate origin, and each corresponding point is symmetrical about the origin,
The change characteristic of vertex coordinates is that the abscissa and ordinate are the original opposite numbers,
So the answer is: abscissa, ordinate are the original opposite number

In the triangle ABC, we know a (1,3, - 5). B (3, - 2,7). If the center of gravity of the triangle is at the origin, then the coordinates of vertex C are?

Knowledge: if the coordinates of the three vertices of a triangle are a (x1, Y1, z1), B (X2, Y2, Z2), C (X3, Y3, Z3), then the coordinates of the center of gravity g are g ((x1 + x2 + x3) / 3, (Y1 + Y2 + Y3) / 3, (z1 + Z2 + Z3) / 3) let C (x, y, z) then the center of gravity g ((x + 1 + 3) / 3, (y + 3-2) / 3, (Z + 7-5) / 3) because the center of gravity is at the origin, so

The edge length of the equilateral triangle ABC is 2, the vertex is at the coordinate origin, and the coordinates of a, B and C points are obtained on the X axis

If point a is at the coordinate origin, then a (0,0)
∵ B is on the x-axis,
∴AB=2,
The coordinate B is (2,0) or (- 2,0)
As CD ⊥ AB in D, then ad = 1 / 2Ab = 1, CD = √ 3,
When the B coordinate is (2,0),
The coordinates of point C are (1, √ 3) or (1, √ 3)
When the B coordinate is (- 2,0),
The coordinates of point C are (- 1, √ 3) or (- 1, √ 3)

The vertex coordinates of the triangle ABC are divided into a (0,0), B (2,0), C (2 roots, 2,2 roots). Rotate the triangle ABC about the origin by 135 degrees in a counter clockwise direction to obtain the triangle a, b'c ', then the coordinates of B' are () and the coordinates of C 'are ()

∵ CX = CY = 2 √ 2, ᙽ AC | 4, (2) the angle between AC and x-axis is 45 °
ν AC 'is aligned with the X axis (135 ° + 45 ℃) = 180 ℃  C' = (- 4,0)
∵|| ab | = 2, point B is on the X axis,  ab 'is at an angle of 135 ° with the X axis, and ᙽ B' is on the bisector of the second quadrant angle
∴B'=(-2/√2,2/√2)=(-√2,√2)

It is known that in the plane rectangular coordinate system, O is the coordinate origin, and the three vertices of △ ABC are a (0,8), B (7,1), C (- 2,1) 1. Find the size of the inner angle B of △ ABC; 2. Let the moving point P satisfy that the vector OP is perpendicular to the vector OC, and find the minimum value of vector PA multiplied by vector Pb

1: Calculate the angle between the vector AB and AC and directly bring it into the formula. Or draw a graph. The tangent value of the inner angle B = 7 / 7 = 1, so B = 45 degrees
2: Let P point (x, y), Op vertical OC can get - 2x + y = 0
PA × Pb yields (- x, 8-y) * (7-x, 1-y) = x ^ 2-7x + y ^ 2-9y + 8 = min
By simplifying the above two formulas, we can get an equation about X
min= 5x^2-25x+8
Obviously, the above is a parabola with an upward start. There is a minimum value. Calculate the minimum value of the parabola. It seems that it is 21.25. Calculate it slowly