In the figure KE is parallel to LF which is parallel to MG which is parallel to NH etc
Also DK:KL= 2:1
DL:LM= 2:1
DM:MN= 2:1 etc
If KE= 2cm,calculate the length of the 10th parallel line,rounded off to one decimal. (N.B NH is the 4th parallel line)
Given that DK:KL = 2:1, DL:LM = 2:1, DM:MN = 2:1, etc., and using the fact that KE is 2 cm, we can determine the length of each parallel line.
Since DK:KL = 2:1, we can let DK be 2x and KL be x.
So, DK = 2x and KL = x.
Similarly, we can let DL be 2y and LM be y, where DL:LM = 2:1.
So, DL = 2y and LM = y.
Using the same logic, we can let DM be 2z and MN be z, where DM:MN = 2:1.
So, DM = 2z and MN = z.
We can continue this pattern to find the ratios of the subsequent parallel lines.
Now, let's find the length of NH, the 4th parallel line.
Since NH is the 4th parallel line, if KE = 2 cm, then KH = 4 cm (twice the number of parallel lines plus the starting length - 2 x 10 = 20, and 20 + 2 = 22, but since NH is the 4th parallel line, we only need to consider 4 copies of this pattern).
Now, to find the length of the 10th parallel line, we can continue this pattern:
KH = 4 cm
IJ = 6 cm
FG = 8 cm
EF = 10 cm
CD = 12 cm
AB = 14 cm
ZX = 16 cm
YX = 18 cm
So, the length of the 10th parallel line is 18 cm when KE is 2 cm.
To solve this problem, we can use the given ratios to determine the lengths of each segment along the parallel lines. Let's start by finding the lengths of the segments on the fourth parallel line, NH.
Given:
DK:KL = 2:1
DL:LM = 2:1
Let's assign variables to the lengths of the segments:
DK = 2x
KL = x
DL = 2y
LM = y
Since we have DK:KL = 2:1, we can write the equation:
2x/x = 2/1
Simplifying the equation, we get:
2x = 2
x = 1
Now, let's use this value to find the length of DL:
DL = 2y = 2(1) = 2
Since NH is the fourth parallel line, we can assume that the lengths of the segments on the first parallel line are half the lengths of the corresponding segments on the fourth parallel line. Therefore, on the first parallel line:
DK = 2x/2 = x
KL = x/2 = 1/2
DL = 2y/2 = y
LM = y/2 = 1/2
Now, let's calculate the length of the 10th parallel line using a pattern we observe. We notice that the lengths of the segments on each parallel line follow a pattern:
1st parallel line: DK, KL
2nd parallel line: DL, LM
3rd parallel line: DM, MN
4th parallel line: DN, NP
5th parallel line: DO, OQ
...
The pattern is that the lengths of the segments alternate between being divided by 2 (e.g., DK, DL, DM,...) and being multiplied by 2 (e.g., KL, LM, MN,...). Therefore, on the 10th parallel line:
DK = (x/2)^10 = (1/2)^10 = 1/1024
KL = (x*2)^9 = (1*2)^9 = 512
So, the length of the 10th parallel line is KL = 512 cm (rounded to one decimal place).
Note: It is important to verify the pattern and check for any additional information or assumptions that may affect the solution.