F.3 to F.4 mathematics(HELP ME!!)

[匿名]

2007-08-29 7:18 pm

1)The vertices of △ABC are A( 3 , -1 ) ,B( -3 , 5 ) and C( 2 , 4 ).Find the coordinates of the orthcenter of △ABC.

2)The distance between the points ( x , y ) and ( -4 , 2 ) is 5 .Find the equation connecting x and y .What kind of graph does this equation represent?

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回答 (1)

用戶22404

2007-08-29 10:43 pm

✔ 最佳答案

1) The equation of line joining points A & B:
[y-(-1)]/(x-3) = [5-(-1)]/(-3-3)
(y+1)/(x-3) = -1
y+1 = -x+3
y=-x+2
Hence, the slope of the altitude perpendicular to line AB is (-1)/(-1) = 1
Therefore, equation of altitude joining vertex C with slope=1 :
(y-4)/(x-2) = 1, y-4=x-2, y=x+2 -----(a)

The equation of line joining points B & C:

[(y-4)/(x-2)]=[(5-4)/(-3-2)]
(y-4)/(x-2)=1/-5
-5(y-4)=x-2
-5y+20=x-2
5y=-x+22
y=-x/5+22/5

Hence, the slope of altitude perpendicular to line BC is -1/(-1/5) = 5
equation of that altitude joining vertex A with slope=5:
[y-(-1)]/(x-3) = 5, y+1=5x-15, hence, y=5x-16-------(b)

(a) =(b):

x+2=5x-16
4x=18
x=4.5-----(c)

Put (c) into (a)
y=4.5+2=6.5

Coordinates of the orthcenter = ( 4.5, 6.5)

2) (y-2)^2 + [x-(-4)]^2 = 5^2
y^2 -4y +4 + x^2 + 8x + 16 = 25
y^2 + x^2 +8x -4y -5 = 0

Circle

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