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A mathematical problem of geometric moving point of unary function in junior middle school
As shown in the figure, the straight line Y=2X+4 intersects the X axis and Y axis at point C and point A respectively; The coordinate of point B is (4,0). When point B passes, BD is perpendicular to AC at point D, and BD intersects OA at point H. ..

(1) Request the analytical formula of BD.

(2) There are two moving points P and Q moving from point C and point O in the positive direction of X axis at the speed of 2 units per second and 1 unit per second respectively. Let the area of triangle PQD be S, point P, and the movement time of point Q be t seconds, and request the functional relationship between S and T (please write the range of corresponding independent variable T directly).

(3) What is the value of t? The area of triangle PQD is 1 of the area of triangle BCD.

(1) Analysis: ∫ straight line Y=2X+4 and X axis, and Y axis intersects with point C and point A respectively; The coordinate of point B is (4,0).

∴A(0,4),B(4,0),C(-2,0)

Through point B, BD is perpendicular to AC at point D, and BD intersects OA at point H.

∴BD slope k =-1/2; The analytical formula of BD: y =-1/2 (x-4) = 2-1/2x.

(2) Analysis: From the meaning of the question: the coordinates of point P: (2t-2,0); Q point coordinates: (t, 0);

D(-4/5, 12/5) is obtained by simultaneous solution of AC and BD analytical expressions.

∴s= 1/2*|t-(2t-2)|* 12/5=6/5|2-t|

The value range of t t >; 0

(3) analysis: s (⊿ BCD) =1/2 * 6 *12/5 = 36/5 = > s (⊿ BCD)/6 = 6/5.

6/5 | 2-t | = 6/5 = = > | 2-t | =1= > t =1or t=3.