As shown in the figure, in the plane rectangular coordinate system, the coordinates of the three vertices of △ ABC are a (2, 3), B (2, 1), C (3, 2) (1) Judge the shape of △ ABC; (2) If △ ABC is rotated one circle along the line of edge AC, the volume of the rotating body is obtained

(1) Answer: triangle is isosceles right triangle;
From the coordinates of points a, B and C,
AC=
(2-3)2+(3-2)2=
2，
BC=
(3-2)2+(2-1)2=
2，
AB=3-1=2，
Because（
2）2+（
2) 2 = 4 = 22, namely ac2 + BC2 = AB2, AC = BC,
Therefore, the triangle is an isosceles right triangle;
(2) The volume of the cone is 1
3π•BC2•AC=1
3π×（
2）2×
2=2
Three
2π．

As shown in the figure, in the plane rectangular coordinate system, the coordinates of the three vertices of the triangle ABC are a (3,4), B (1,2) C (5,2) (1) to find the length of ab (2) Judge triangle ABC

AB = root ((3-1) square + (4-2) square) = root 8 = 2 root sign 2
Similarly, AC = 2 radical sign 2
BC=2
So isosceles right triangle

As shown in the figure, in the plane rectangular coordinate system, the coordinates of each vertex of △ ABC are a (0,1), B (2,0), C (2,1.5) (1). Find the area of △ ABC 2 if there is a point P (a, 2 / 1) in the second quadrant, the area of the quadrilateral abop is represented by an expression containing a 3 under the condition of 2, is there a point P that makes the area of the quadrilateral abop equal to the area of △ ABC? If so, find out the coordinates of point P. if not, please explain the reason

(1) By a (0,1) B (2,0) C (2,1.5)
In other words, BC = 1.5, distance from a to BC = 2,
∴S△ABC=1.5×2÷2=1.5.
(2) Let P (a, 1 / 2)
Abop area of quadrilateral s = △ AOB area + △ AOP area
=1×2÷2+1×（-a）÷2
=1-a/2.
(3) Let 1-A / 2 = 1.5
A = - 1., P (- 1,1 / 2)

It is known that the coordinates of the three vertices of △ ABC are a (2,2), B (4,2), C (6,4). Taking the origin as the similitude center, the △ ABC is reduced to obtain △ def If the ratio between it and the corresponding side of △ ABC is 1:2, what is the coordinates of the corresponding point after the transformation of midpoint P of line AC

For example, D (1,1) or (- 1, - 1)

The coordinates of the three vertices of △ ABC are a (2,2) B (4,2) C (6,4). The hospital point O is the similar center, which reduces △ ABC Is the ratio of the corresponding edge of △ a'b'c 'and △ ABC obtained after transformation is 1:2, and the coordinates of point a'b'c' are obtained

A'(1,1) B'(2,1) C'(3,2)

As shown in the figure, the coordinates of the three vertices of △ ABC are a (- 2,6), B (6, - 2), C (- 4, - 4). Take the origin o as the similitude center, reduce △ ABC so that the ratio of △ def to the corresponding edge of △ ABC is 1:2, and then calculate the coordinates of each vertex of △ def

∵A（-2，6），B（6，-2），C（-4，-4），
The similarity ratio is 1
2. If △ ABC is reduced, the coordinates of its corresponding vertices are (- 1,3), (3, - 1), (- 2, - 2),
As shown in the figure:

If we rotate the coordinates of △ B 'around the origin, then we get the coordinates of △ B (2,2' C) of ABC

Because C (2 √ 2,2 √ 2), according to the Pythagorean theorem, AC length is 4, and AC is exactly at the straight line of the angular bisector of the first quadrant

As shown in the figure, in the plane rectangular coordinate system, the three vertices of △ ABC are a (1,2) B (3,2) C (2,3). Try to judge the shape of △ ABC

AB²=(3-1)²+(2-2)²=4
BC²=(2-3)²+(3-2)²=2
CA²=(2-1)²+(3-2)²=2
BC=CA BC²+CA²=AB²
A triangle is an isosceles right triangle