Show that the mid point of line joining points (-5,12) and (-1,-12) is a point of trisection of the line joining the points (-8,-5) and (7,10).
midpoint of (-5,12) and (-1,-12)
= ( (-5-1)/2 , (12-12)/2 ))
= (-3, 0)
I will use vectors to find the points of trisection
i.e. in the ratio of 1:2
one point = (1/3)(7,10) + (2/3)(-8,-5)
= (7/3, 10/3) + (-16/3 , -10/3)
= (-9/3 , 0)
= (-3,0)
which is the midpoint of above
other point = (1/3)(-5,12) + (1/3)(-1,-12)
= (-5/3 , 4) + (-1/3 , -4)
= (-2 , 0)
This answer helped me
To find the mid-point of a line segment with endpoints (x1, y1) and (x2, y2), we use the following formulas:
Mid-point coordinates = ((x1 + x2)/2, (y1 + y2)/2)
Let's find the mid-point of the line segment joining the points (-5, 12) and (-1, -12):
Mid-point coordinates = ((-5 + -1)/2, (12 + -12)/2)
= (-6/2, 0/2)
= (-3, 0)
Now, we need to find the equation of the line passing through the points (-8, -5) and (7, 10).
First, we find the slope of the line:
m = (y2 - y1) / (x2 - x1)
= (10 - (-5)) / (7 - (-8))
= (10 + 5) / (7 + 8)
= 15 / 15
= 1
The equation of the line in point-slope form is:
y - y1 = m(x - x1)
Using the point (-8, -5):
y - (-5) = 1(x - (-8))
y + 5 = x + 8
y = x + 8 - 5
y = x + 3
Now, we need to find the point of trisection of this line.
Let's assume the point of trisection as (x, y).
Using the mid-point formula, we have:
(((-3) + (-8))/2, (0 + (-5))/2) = (x, y)
Simplifying:
(-11/2, -5/2) = (x, y)
Therefore, the point (-11/2, -5/2) is the point of trisection of the line joining the points (-8, -5) and (7, 10).
To show that the midpoint of the line joining points (-5,12) and (-1,-12) is a point of trisection of the line joining the points (-8,-5) and (7,10), we need to find the coordinates of the midpoint of the first line and check if it divides the second line into two equal segments.
First, let's find the midpoint of the line joining (-5,12) and (-1,-12). The midpoint formula is given by:
Midpoint = ((x1 + x2)/2, (y1 + y2)/2)
Plugging in the values, we have:
Midpoint = ((-5 + -1)/2, (12 + -12)/2)
= (-6/2, 0)
= (-3, 0)
So, the midpoint of the first line is (-3, 0).
Next, let's find the length of the line segment between the points (-8,-5) and (-3, 0), as well as the length of the line segment between the points (-3, 0) and (7, 10).
Using the distance formula:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
For the line segment between (-8,-5) and (-3, 0), we have:
Distance = sqrt((-3 - (-8))^2 + (0 - (-5))^2)
= sqrt((5)^2 + (5)^2)
= sqrt(50)
= 5 * sqrt(2)
For the line segment between (-3, 0) and (7, 10), we have:
Distance = sqrt((7 - (-3))^2 + (10 - 0)^2)
= sqrt((10)^2 + (10)^2)
= sqrt(200)
= 10 * sqrt(2)
Notice that the ratio of the lengths of the line segments is:
(5 * sqrt(2)) / (10 * sqrt(2)) = 1/2
Since the ratio is 1/2, we can conclude that the midpoint of the line joining (-5,12) and (-1,-12), which is (-3, 0), is indeed a point of trisection of the line joining the points (-8,-5) and (7,10).