The image of the first order function y = (- radical 3 / 3) x + 1 intersects with X axis and Y axis respectively at two points a and B. taking line AB as the edge, an equilateral triangle ABC is made in the first quadrant (1) Find the area of ABC (2) if there is a point P (a, 1 / 2) in the second quadrant, try the formula containing a to represent the area of quadrilateral ABPO, and calculate the value of a when the area of triangle ABP is equal to that of triangle ABC

The image of the first order function y = (- radical 3 / 3) x + 1 intersects with X axis and Y axis respectively at two points a and B. taking line AB as the edge, an equilateral triangle ABC is made in the first quadrant (1) Find the area of ABC (2) if there is a point P (a, 1 / 2) in the second quadrant, try the formula containing a to represent the area of quadrilateral ABPO, and calculate the value of a when the area of triangle ABP is equal to that of triangle ABC

(1) a (radical 3.0) B (0.1) so AB = 2 so equilateral triangle ABC = 2x1 / 2x radical 3 = radical 3 (2) s quadrilateral ABPO = s triangle AOB + s triangle BOP

It is known that AB is parallel to y = - 3 / 3 root sign 3x, and intersects B (0,1) with y axis, and intersects a with X axis, and makes equilateral △ ABC with segment AB in the first quadrant Ask for: (1) AB analytic formula (2) Coordinates of point C (3) The first term line P (m, 0.5), and s △ ABP = s △ ABC, find P coordinate

(1) AB analytic formula y = - 3 / 3 root 3 x + 1
B (radical 3,0)
(2) For example, Bao = 30 °, cab = 60 °, CA ⊥ X axis, and ab = 2
Then CA = 2, C (radical 3,2)
(3) S △ ABP = s △ ABC knows the distance from C to ab = the distance from P to ab
The distance from C to ab = root 3, P to ab = (- 3 / 3 root 3 M + 0.5) absolute value / (2 / root 3) = 2
M = - 3 root sign 3 / 2 or M = 5 root number 3 / 2
P (- 3 root number 3 / 2,0.5) or P (5 root number 3 / 2,0.5)

As shown in the figure, the straight line y = - radical 3 / 3x + 1 intersects x-axis and y-axis respectively, finds the coordinates of two points B and a (1) turns the triangle line AOB with the line AB as the axis, The straight line y = negative root 3x + root 3 and Y axis, X axis respectively intersect two points of A. B. (2) if the triangle AOB is turned over along the straight line, point 0 falls at point C on the plane, and an equilateral triangle BCD is made with BC as one side, and the coordinates of point D are calculated.

Angle ABO = arctan (root 3 / 1) = 60 = angle ABC = angle CBD
Angle CBD = 180 - angle OBC, D falls on X axis
BC=BO=1
BD = 2 * cos (angle CBD) * BC = 1 (or angle CBD = 60, BD = BC = 1 (equilateral triangle BCD))
OD=OB+DB=2
D=(2,0)

As shown in the figure y = (root 3) / 3x + B, passing through point B (- radical 3,2) and intersecting point a with X axis, the parabola y = 1 / 3x square is shifted left and right along the X axis, and then the throwing is obtained (3) In the process of y = 1 / 3x square translation, the triangle PAB is turned along the straight line AB to get the triangle DAB. Can point d fall on the parabola C? If it can find out the vertex P coordinates of the parabola C at this time? If not, why not?

1.y=√3/3 x+b,2=√3/3(-√3)+b,b=3, ∴y=√3/3 x+3,
tan∠BAO=√3/3,∠BAO=30°, ∵ ∴
2. After the parabola y = 1 / 3x ^ 2 is translated, the parabola is y = 1 / 3 (x-a) ^ 2, which intersects with the Y axis at e (0,1 / 3A ^ 2), EF ∥ x,  f (x1,1 / 3A ^ 2), 1 / 3A ^ 2 = = √ 3 / 3 X1 + 3,
1 / 3A ^ 2 = = 1 / 3 (x1-a) ^ 2, a = - √ 3, or a = 3 √ 3, parabolic C: y = 1 / 3 (x + √ 3) ^ 2, or y = 1 / 3 (x-3 √ 3) ^ 2
3. On y = 1 / 3 (x-a) ^ 2, P (a, 0), fold along the line AB to get point d (x1, Y1), PD midpoint, on the line AB, and KPD = - √ 3, Y1 / 2 = = √ 3 / 3 (x1 + A) / 2 + 3, Y1 / (x1-a) = - √ 3, Y1 = 1 / 3 (x1-a) ^ 2, and get a = 0. In the translation process, the triangle PAB is turned along the line AB to get the triangle DAB, and the point d does not fall on the parabola C

The line L1: y = - radical 3x + radical 3 intersects point a and B with X axis and Y axis respectively, △ AOB and △ ACB are symmetrical about the line L, and the coordinates of point C are obtained

Δ AOB and △ ACB are symmetric about a line L
So C and o are symmetric about L
So the slope of the line OC is √ 3
Passing point O (0,0)
So OC y = √ 3
c（x,√3x）
So the midpoint of OC is on ab
That is (x / 2, (√ 3x / 2)) on ab
That is √ 3x / 2 = - √ 3 * x / 2 + √ 3
X=1
c(1,√3)

The straight line L: y = - radical 3x + radical 3 intersects point a and B with X axis and Y axis respectively, △ AOB and △ ACB are symmetrical about the line L, and the analytic formula of straight line passing through point B and C is obtained

l: Y = - √ 3 (x-1), the coordinates of point B are (0, √ 3), and the coordinates of point a are (1,0)
Therefore ∠ ABO = 30 ° and △ AOB and △ ACB are symmetric about the straight line L, so ∠ ABC = ∠ ABO = 30 ° is obtained, so ∠ OBC = 60 ° and the included angle between the straight line BC and the positive direction of x-axis is 150 ° so the analytical formula is
Y - √ 3 = - X / √ 3, i.e
y=-x/√3 + √3

Root sign (x + Y-8) + root sign (8-x-y) = root sign (3x-y-a) + root sign (x-2y + A + 3), can three line segments with length of X, y, a form a triangle? Good extra points,

According to the meaning of the title, x + Y-8 ≥ 0,
And 8 - (x + y) ≥ 0, that is, x + Y-8 ≤ 0
So x + Y-8 = 0,
So x + y = 8
So root sign (3x-y-a) + radical sign (x-2y + A + 3) = 0
Because the root sign (3x-y-a) ≥ 0, the root sign (x-2y + A + 3) ≥ 0
So 3x-y-a = 0
x-2y+a+3=0 ③
It can be solved by ①, ② and ③
x=3,y=5,a=4
So you can make a triangle, and it's a right triangle, and the angle that the Y side faces is a right angle
It's very hard to fight. I don't know if you understand. It mainly uses the implied condition that the number in the root is greater than or equal to 0

As shown in the figure, the straight line y = - root 3x + 4, root sign 3 is relative to point a with X axis, and point p.3 is compared with line y = root sign 3x, and the moving point e starts from origin o, with the order of 1 / s 3. Starting from the origin o, the moving point e extends o at a speed of 1 unit per second_ P_ They are not perpendicular to the point a and e of the line, which are perpendicular to the point a and point E. Let the area of overlapped part of rectangular ebof and triangle OPA be s when moving for T seconds, and find the functional relationship between S and t.

Find P (2,2 √ 3) from the first two questions
Delta OPA is an equilateral triangle
OP=OA=PA=4
3. When 0

As shown in the figure, the three semicircles are circumscribed in turn, and their centers are on the x-axis and are connected with the straight line y= Three If the radii of the three semicircles are R1, R2 and R3 in turn, then when R1 = 1, R3=______ .

Three semicircles are tangent to the line y = 33x in turn and their centers are on the x-axis, ∵ the inclination angle of the line y = 33x is 30 °, oo1 = 2r1002 = 2r2 = oo1 + R1 + R2 = 3R1 + R2003 = 2r3,  2r2 = 3R1 + R2,

It is known that, as shown in the figure, the straight line y = - radical 3x + 2, root sign 3 and X axis and Y axis intersect at point a and point B respectively, D is a point on Y axis. If the triangle DAB is folded along the line Da, point B just falls at point C on the positive half axis of X axis, and the analytic formula of line CD is obtained

y=-√3x+2√3
The coordinates of point a (2,0) and point B (0,2 √ 3) are obtained
The triangle DAB folds along the line da
So AB = AC, DB = DC
AB＝√〔(2√3)^2+2^2〕=4
AC = 4, so the coordinates of point C are (4,0)
Let the coordinates of point d be (0, y)
BD=2√3+OD=DC
DC^2=OC^2+OD^2
(2√3+OD)^2=4^2+OD^2
Od = √ 3 / 3,
OD=|y|=√3/3,y=±√3/3
It can be seen from the question that point B just falls at point C on the positive half axis of the x-axis
So point D can only be on the lower half axis of Y axis, that is, the coordinates of point D are (0, - √ 3 / 3)
Let the analytic formula of CD be: y = KX + B
Substituting the coordinates of point C as (4,0) and the coordinates of point D as (0, - √ 3 / 3) into the analytic formula, we can get
0=4k+b
－√3/3=b
The solution is: k = √ 3 / 12, B = - √ 3 / 3
Let the analytic formula of CD be: y = (√ 3 / 12) x - √ 3 / 3