The points A(-1,-2) B(6,2) and C(4,6) form a triangle ABC. M is the mid-point of AB and N is the mid-point of AC.
Prove that MN is parallel to BC and equal in length to half the length of BC.
midpoint M is ( (-1+6)/2 , (-2+2)/2 ) = ( 5/2 , 0)
midpoint N is ( (-1+4)/2 , (-2+6)/2 ) = ( 3/2 , 2)
slope BC = (6-2)/(4-6) = 4/-2 = -2
slope of MN - (0-2)/(5/2 - 3/2) = -2
So BC is || to MN
length of BC = √( (6-2)^2 + (4-6)^2) = √(16+4)
= √20 = 2√5
length of MN = √( (0-2)^2 + (5/2-3/2)^2
= √( 4+1) = √5
which is 1/2 of BC
To prove that MN is parallel to BC and equal in length to half the length of BC, we can make use of the properties of midpoints in a triangle.
Step 1: Find the coordinates of the midpoints M and N.
To find the midpoint of a line segment, we use the midpoint formula:
Midpoint (Mx, My) = [(Ax + Bx) / 2, (Ay + By) / 2]
Midpoint (Nx, Ny) = [(Ax + Cx) / 2, (Ay + Cy) / 2]
Substituting the coordinates of A, B, and C into the formulas, we get:
Midpoint M = [(-1 + 6) / 2, (-2 + 2) / 2] = [2.5, 0]
Midpoint N = [(-1 + 4) / 2, (-2 + 6) / 2] = [1.5, 2]
So, the coordinates of M are (2.5, 0), and the coordinates of N are (1.5, 2).
Step 2: Determine the inclination of BC.
To determine the inclination (slope) of a line, we use the formula:
Slope (m) = (By - Ay) / (Bx - Ax)
Substituting the coordinates of B and C into the formula, we get:
Slope BC (m) = (2 - 6) / (6 - 4) = -4 / 2 = -2
Step 3: Determine the inclination of MN.
Since M is the midpoint of AB, the line segment MN is parallel to BC. Therefore, it will have the same slope as BC.
Slope MN (m) = Slope BC = -2
Step 4: Determine the length of BC.
To find the length of a line segment, we use the distance formula:
Distance BC = sqrt((Bx - Cx)^2 + (By - Cy)^2)
Substituting the coordinates of B and C into the formula, we get:
Distance BC = sqrt((6 - 4)^2 + (2 - 6)^2) = sqrt(2^2 + (-4)^2) = sqrt(4 + 16) = sqrt(20) = 2sqrt(5)
Step 5: Determine the length of MN.
To find the length of a line segment, we use the distance formula:
Distance MN = sqrt((Mx - Nx)^2 + (My - Ny)^2)
Substituting the coordinates of M and N into the formula, we get:
Distance MN = sqrt((2.5 - 1.5)^2 + (0 - 2)^2) = sqrt(1^2 + (-2)^2) = sqrt(1 + 4) = sqrt(5)
Step 6: Compare the inclination and length of MN with BC.
We have determined that the slope of MN is -2, which is the same as the slope of BC. This shows that MN is parallel to BC.
We have also found that the length of MN is sqrt(5), which is equal to half the length of BC, which is 2sqrt(5) / 2 = sqrt(5).
Therefore, we have proven that MN is parallel to BC and equal in length to half the length of BC.