Answer:
y = 6
Explanation:
You want the value of y that makes segment UV ⊥ WX, given points U(-6, 7), V(1, -7), W(6, 4), and X(10, y).
Perpendicular
The lines will be perpendicular when the dot product of the direction vectors is zero.
The direction of UV is V -U = (1, -7) -(-6, 7) = (1 +6, -7-7) = (7, -14). We can simplify this by dividing by 7 to get ...
direction UV = (1, -2)
The direction of WX is X -W = (10, y) -(6, 4) = (4, y-4).
Dot product
The dot product of two vectors is the sum of products of their components:
UV•WX = (1, -2)•(4, y-4) = (1)(4) +(-2)(y -4) = 4 -2y +8 = 12 -2y
We want this to be zero, so ...
12 -2y = 0
2y = 12
y = 6
The value of y is 6.
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Additional comment
As we found when computing the direction of UV, its slope is -2. That means the slope of WX will be -1/(-2) = 1/2. The equation of line WX can be written in point-slope form as ...
y -4 = 1/2(x -6)
Using x = 10, we can solve for y as ...
y = 1/2(10 -6) +4 = 2+4 = 6
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