the three vertices of a triangle are at A (-4,5), B(-5,3) and C (3,-4), show that the line joining the midpoint of the sides AC and BC is parallel and equal to one-half of AB . solve
AB has slope 2 and length √5
mid(AC) = (-1/2,1/2)
mid(BC) = (-1,-1/2)
so that segment has slope 2 and length √5/2
To solve this problem, we need to follow a few steps:
Step 1: Find the midpoint of sides AC and BC.
Step 2: Determine the equation of the line passing through these midpoints.
Step 3: Calculate the length of side AB.
Step 4: Verify if the line's slope is equal to one-half the length of AB.
Let's work through each step:
Step 1: Find the midpoint of sides AC and BC.
The midpoint of a line segment between two points (x1, y1) and (x2, y2) is given by the coordinates:
midpoint = ((x1 + x2)/2, (y1 + y2)/2)
For side AC:
Midpoint of AC = ((-4 + 3)/2, (5 - 4)/2)
= (-1/2, 1/2)
For side BC:
Midpoint of BC = ((-5 + 3)/2, (3 - 4)/2)
= (-1, -1/2)
So, the midpoint of AC is (-1/2, 1/2) and the midpoint of BC is (-1, -1/2).
Step 2: Determine the equation of the line passing through these midpoints.
We can use the two-point form of a line which is given by the equation:
y - y1 = m(x - x1)
Where (x1, y1) is a point on the line, and m is the slope of the line.
Using the midpoint (-1/2, 1/2) of AC, and the midpoint (-1, -1/2) of BC, we can calculate the slope using the formula:
m = (y2 - y1)/(x2 - x1)
m = (1/2 - (-1/2))/(-1 - (-1/2))
m = (1/2 + 1/2)/(-1 + 1/2)
m = (1)/(-1/2)
m = -2
The equation of the line passing through the midpoints is:
y - 1/2 = -2(x + 1/2)
Simplifying this equation, we get:
y = -2x
So, the equation of the line passing through the midpoints is y = -2x.
Step 3: Calculate the length of side AB.
The length of side AB can be found using the distance formula, given by:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
For side AB, let's consider points A (-4, 5) and B (-5, 3):
d = sqrt((-5 - (-4))^2 + (3 - 5)^2)
d = sqrt((-5 + 4)^2 + (-2)^2)
d = sqrt((-1)^2 + 4)
d = sqrt(1 + 4)
d = sqrt(5)
So, the length of side AB is sqrt(5).
Step 4: Verify if the line's slope is equal to one-half the length of AB.
The equation of the line passing through the midpoints is y = -2x, which has a slope of -2.
One-half the length of AB is (1/2) * sqrt(5).
To verify if the line's slope is equal to one-half the length of AB, we need to compare the two values:
-2 (slope of the line) = (1/2) * sqrt(5) (one-half the length of AB)
Simplifying this equation, we get:
-2 = (1/2) * sqrt(5)
-4 = sqrt(5) (by multiplying both sides by 2)
16 = 5 (by squaring both sides)
This equation is not true. Therefore, the line joining the midpoint of sides AC and BC is not parallel and equal to one-half of AB.
Hence, we have proved that the line joining the midpoint of the sides AC and BC is not parallel and equal to one-half of AB.