Generative synthesis of junior high school mathematics

1, (Suqian 05) known: As shown in the figure, in △ABC, ∠ c = 90, AC = 3cm, CB = 4cm. The two moving points p and q move clockwise along the edge of △ABC from point A and point C, respectively. When point Q moves to point A, the movement of point P and point Q stops.

(1) When the time is what value, the area of the triangle (shaded part in the figure) with P, C and Q as vertices is equal to 2cm2.

(2) When point P and point Q move, the shape of the shadow part also changes. Let the shaded area surrounded by PQ and △ABC be s (cm 2), find out the functional relationship between S and time, and point out the range of independent variables.

(3) Is there a maximum shadow area s during the movement of points P and Q? If yes, request the maximum value; If not, please explain why.

Analysis: This is a translation problem to the moving point. With the movement of the moving points P and Q, the shape of the shadow part changes from triangle to quadrilateral, then to triangle, and the area of the shadow part also changes accordingly.

However, the problem 1 can be fixed as the area of the figure 1- 1 under static conditions;

Question 2 should pay attention to three critical States: =2, =3, =4.5, so

It should be discussed in three situations: 0 < ≤ 2, 2 < ≤ 3 and 3 < ≤ 4.5.

Problem 3 only needs to be transformed into finding the extreme value of three analytical expressions of problem 2 and comparing them.

Solution: (1) s △ pcq = PC? 6? 1cq = = 2, the solution is = 1, = 2.

When the time is 1s or 2s, ∴, s △ pcq = 2cm2;

(2)① When 0 < ≤ 2, s = =

② When 2 < ≤ 3, s = =

③ When 3 < ≤ 4.5, s = =

(3) yes; (1) when 0 < ≤ 2, when =, s has the maximum value, s1=;

(2) When 2 < ≤ 3, when = 3, S has the maximum value, S2 =；;

③ When 3 < ≤ 4.5, when =, s has the maximum value, S3 =；;