What are the lengths of the segments into which the y-axis divides the segment joining (-6,-6) and (3,6) ?
make a sketch
label the intersection withe the y-axis as (0,y)
I have two similar triangles and by ratios
(6-y)/3 = (6+y)/6
18+3y = 36-6y
9y=18
y=2
so the sides of the smaller right-angled triangle are 3 and 4, and I recognize the 3, 4, 5 triangle
the other triangle is 6 , 8, 10
So the length of the two segments you want are 10 and 5.
Long way:
Find the equation of the line joining the two points.
Find the y-intercept of the line
Find the length of the two segments using the distance formula between two points
To find the lengths of the segments into which the y-axis divides the segment joining (-6, -6) and (3, 6), we need to consider the points where the line intersects the y-axis.
The y-axis is a vertical line passing through the origin, so its equation is x = 0. To find the intersection points of this line with the given segment, we substitute x = 0 into the equation of the segment and solve for y.
Let's find the equation of the line passing through the points (-6, -6) and (3, 6) using the slope-intercept form, y = mx + b.
First, find the slope (m) using the formula:
m = (change in y)/(change in x) = (6 - (-6))/(3 - (-6)) = 12/9 = 4/3.
Now, substitute one of the points and the slope into the equation to solve for b:
-6 = (4/3)(-6) + b
-6 = -8 + b
b = -6 + 8
b = 2.
Therefore, the equation of the line passing through the points (-6, -6) and (3, 6) is y = (4/3)x + 2.
Next, substitute x = 0 into the equation to find the y-intercept:
y = (4/3)(0) + 2
y = 2.
This tells us that the line intersects the y-axis at the point (0, 2).
Now we have two line segment lengths:
1. The segment from (-6, -6) to (0, 2).
2. The segment from (0, 2) to (3, 6).
To find the lengths of these segments, we use the distance formula:
distance between two points (x1, y1) and (x2, y2) = sqrt( (x2 - x1)^2 + (y2 - y1)^2 ).
Let's calculate the lengths of the segments:
1. Length of the segment from (-6, -6) to (0, 2):
distance = sqrt( (0 - (-6))^2 + (2 - (-6))^2 )
= sqrt( (6)^2 + (8)^2 )
= sqrt( 36 + 64 )
= sqrt( 100 )
= 10.
Therefore, the length of this segment is 10.
2. Length of the segment from (0, 2) to (3, 6):
distance = sqrt( (3 - 0)^2 + (6 - 2)^2 )
= sqrt( (3)^2 + (4)^2 )
= sqrt( 9 + 16 )
= sqrt( 25 )
= 5.
Therefore, the length of this segment is 5.
In conclusion, the y-axis divides the segment joining (-6, -6) and (3, 6) into two segments: one with a length of 10 units and the other with a length of 5 units.