Find the length of the median of a trapezoid with vertices at C(3,1) D(10,1) E(7,9) and F(5,9)
You find the midpoint Mid point = ( x+x/2,y+y/2) then you use the distance formula to find the length
To find the length of the median of a trapezoid, we first need to identify which side of the trapezoid is the base and which side is the top.
In this case, the trapezoid has two parallel sides: CD and EF. Let's assume that CD is the base and EF is the top.
The median of a trapezoid is a line segment connecting the midpoints of the non-parallel sides. In this case, the non-parallel sides are CE and DF.
To find the length of the median, we need to calculate the distance between the midpoints of CE and DF.
Step 1: Find the midpoint of CE
The midpoint of CE is calculated by taking the average of the x-coordinates and the average of the y-coordinates of C and E.
The x-coordinate of the midpoint is:
(x-coordinate of C + x-coordinate of E) / 2 = (3 + 7) / 2 = 10 / 2 = 5
The y-coordinate of the midpoint is:
(y-coordinate of C + y-coordinate of E) / 2 = (1 + 9) / 2 = 10 / 2 = 5
Therefore, the midpoint of CE is M1(5, 5).
Step 2: Find the midpoint of DF
The midpoint of DF is calculated in the same way as the midpoint of CE.
The x-coordinate of the midpoint is:
(x-coordinate of D + x-coordinate of F) / 2 = (10 + 5) / 2 = 15 / 2 = 7.5
The y-coordinate of the midpoint is:
(y-coordinate of D + y-coordinate of F) / 2 = (1 + 9) / 2 = 10 / 2 = 5
Therefore, the midpoint of DF is M2(7.5, 5).
Step 3: Find the distance between M1 and M2
The distance between two points can be found using the distance formula:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
In this case, we want to find the distance between M1(5, 5) and M2(7.5, 5).
Distance = sqrt((7.5 - 5)^2 + (5 - 5)^2) = sqrt(2.5^2 + 0^2) = sqrt(6.25) = 2.5
Therefore, the length of the median of the trapezoid is 2.5 units.