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Come on, that's awesome. Help! In 2009, the math finale of Tianjin senior high school entrance examination was named
In 2009, there were 26 finale questions in Tianjin senior high school entrance examination: (2) Under the condition of (1), if the two intersections of the function Y 1 and the image of Y2 are A and B, when the area of the triangle ABM is112 cubic, find the value of t;

Note: The coordinate given by M in the question is (t, t).

Question: Why |t-T|=√2h (1)?

The formula (1) can be interpreted in two ways:

One method directly uses the distance formula from point to straight line;

The distance from the point (x 1, y 1) to the straight line ax+by+c=0 is:

D=|ax 1+by 1+c|/ under the quadratic root sign (a 2+b 2)

Therefore, the distance from the point M( t, t) to the straight line y=x, that is, x-y=0 is:

H=d=|t-T|/ Under the square root (12+12) = | t-t |/√ 2.

Therefore |t-T|=√2h (1).

The second method:

Let the straight line passing through point M( t, t) and perpendicular to the straight line Y=X be Y=kX+b and the vertical foot be n.

Then k=- 1

T=-t+b

There is b=T+t

Therefore, a straight line passing through point M( t, t) and perpendicular to line Y=X is y =-x+t+t.

Simultaneous equation: Y = X;;

Y=-X+T+t

X=(T+t)/2。

Y=(T+t)/2

The vertical foot is N((T+t)/2, (T+t)/2).

According to the distance formula between two points

H = Mn = | t-t |/√ 2。

Therefore |t-T|=√2h (1).