In the rectangular coordinate system, the origin of the coordinate is O, a (- 2,4), B (4,2) are known. (1) find the area of △ AOB; (2) find the point P on the x-axis to minimize the value of PA + Pb, and find the coordinate of point P

In the rectangular coordinate system, the origin of the coordinate is O, a (- 2,4), B (4,2) are known. (1) find the area of △ AOB; (2) find the point P on the x-axis to minimize the value of PA + Pb, and find the coordinate of point P

(1) Make AC ⊥ X axis and BD ⊥ X axis respectively through a and B, and make C and D. (1 point) ⊥ AC = 4, BD = 2, CD = 6. (2 points) ⊥ s △ AOB = s trapezoid acdb-s △ AOC-S △ BOD = 12 × (4 + 2) × 6-12 × 2 × 4-12 × 4 × 2 = 10. (4 points) (2) make point E (4, - 2) of B symmetrical with respect to X axis, and connect the intersection X axis of AE to P. (6 points) let the analytical formula of straight line AE be Y = KX + B. ∵ a (- 2,4), e (4, - 2), ｜ 4 = − 2K + B − 2 = 4K + B. the solution is k = − 1b = 2. The analytic expression of ｜ straight line AE is y = - x + 2. (7 points) when y = 0, x = 2. P (- 2,0). (8 points)

In the rectangular coordinate system, the origin of the coordinate is O, a (- 2,4), B (4,2) are known. (1) find the area of △ AOB; (2) find the point P on the x-axis to minimize the value of PA + Pb, and find the coordinate of point P

(1) Make AC ⊥ X axis and BD ⊥ X axis respectively through a and B. make C and D. (1 point) ⊥ AC = 4, BD = 2, CD = 6. (2 points) ⊥ s △ AOB = s trapezoid acdb-s △ AOC-S △ BOD = 12 × (4 + 2) × 6-12 × 2 × 4-12 × 4 × 2 = 10. (4 points) (2) make point E (4, - 2) of B symmetry about X axis

It is known that in the rectangular coordinate system with point o as the origin, if point a (radical 3,0), point B (0,3) and RT △ ABO have a common hypotenuse and an congruent straight line
If the angle triangle and RT △ ABO have a common edge and are congruent, then the unknown vertex coordinates of the right triangle satisfying the condition in the four quadrants (note that it is not calculated on the coordinate axis) are
There are five answers, four of which I found, that is (- (root 3) | 2,3 | 2) do not know, please answer, because the final exam, thank you very much!

The case of common hypotenuse ab
Let this point be p, and PQ ⊥ X axis be set at Q through P,
BAP=∠OBA=30°,∴∠POQ=∠BAO-30°=30°,AP=OB=3,
In RT Δ Apq, PQ = 1 / 2AP = 3 / 2, AQ = 3 √ 3 / 2,
∴OQ=AP-OA=√3/2,
∴P(-√3/2,3/2).

In Cartesian coordinates, we call a point whose abscissa and ordinate are integers integral
In the rectangular coordinate system, we call the points whose abscissa and ordinate are integers the integral points. Moreover, it is stipulated that the interior of the square does not contain the points on the boundary. Observe the square whose center is at the origin and one side is parallel to the x-axis as shown in the figure: there is one integral point in the square whose side length is 1, and there are nine integral points in the square whose side length is 3 , then the number of integral points inside the square with side length of 6 is ()

This is very simple. For a square with a side length of 6, and the center is the origin, you can know that the coordinates are: - 3 ≤ x ≤ 3, and the Y coordinates are - 3 ≤ y ≤ 3, but the subject requires that the boundary cannot be obtained, so the number of integers x can take is: - 2, - 1,0,1,2, 5 in total. Similarly, the number of integers y can take is: - 2, - 1,0,1,2, 5 in total, so the number of integral points in the square with a side length of 6 is 5 * 5, that is 25