In a parallelogram ABCD, diagonals AC & BD intersect at O.
E, F, G, & H be the midpoints of sides AO, DO, CO & BO.
Find the ratio of (EF+GH+HG+EH) TO (AB+BC+CD+AD)
To find the ratio of the sum of lengths of line segments EF, GH, HG, and EH to the sum of lengths of sides AB, BC, CD, and AD, we need to know the relationship between the diagonals AC and BD in a parallelogram.
In a parallelogram, the diagonals bisect each other. This means that the points E and G are the midpoints of diagonal AC, and points F and H are the midpoints of diagonal BD.
To find the ratio, we need to calculate the lengths of the line segments EF, GH, HG, and EH, as well as the lengths of sides AB, BC, CD, and AD.
Let's assume the length of side AB is 'a' and the length of side AD is 'b'. Since it is a parallelogram, the opposite sides are congruent, so the length of side BC is also 'a' and the length of side CD is 'b'.
The length of diagonal AC is the sum of the lengths of sides AD and BC, which is 'a + b'.
Similarly, the length of diagonal BD is the sum of the lengths of sides AB and CD, which is also 'a + b'.
Since E and G are the midpoints of diagonal AC, the length of EG is half the length of diagonal AC, so EG = (a + b)/2.
Similarly, FH = (a + b)/2.
Since F and H are the midpoints of diagonal BD, the length of FH is half the length of diagonal BD, so FH = (a + b)/2.
Now, let's calculate the length of EF and HG. Since E and G are midpoints of diagonal AC, EF = EG = (a + b)/2. Similarly, HG = FH = (a + b)/2.
Lastly, the sum of lengths of EF, GH, HG, and EH is:
EF + GH + HG + EH = (a + b)/2 + (a + b)/2 + (a + b)/2 + (a + b)/2
= 4(a + b)/2
= 2(a + b)
The sum of lengths of sides AB, BC, CD, and AD is:
AB + BC + CD + AD = a + a + b + b
= 2(a + b)
Now, we can find the ratio:
(EF + GH + HG + EH)/(AB + BC + CD + AD) = 2(a + b)/2(a + b)
= 2/2
= 1
Therefore, the ratio of (EF + GH + HG + EH) to (AB + BC + CD + AD) is 1:1, which means they are equal.