with 2 pentagons (A, B, C, D, E) & F, G, H, I, J.) start with A in lower left corner. Same with F. AE=10, BC=8: FG=9, FJ=15. There are diagonal line CE & HJ. x= AB, Y=GH. Find the scale factor of ABCDE to FGHIJ; need value of x & y; perimeter of ABCDE is 38, what is perimeter of FGHIJ; length of HJ=15, CE=? I am trying to help a grand daughter.
To find the scale factor of ABCDE to FGHIJ, we can compare the lengths of corresponding sides. Let's label the scale factor as k.
Given AE = 10 and FG = 9, we can set up the proportion:
AE / FG = AB / GH
10 / 9 = x / (k * y)
To find the value of x, we can rearrange the equation:
x = (10 / 9) * (k * y)
To find the perimeter of ABCDE, we add up the lengths of all five sides:
Perimeter of ABCDE = AB + BC + CD + DE + EA = x + 8 + CE + 10 + x
Since the perimeter is given as 38, we can set up the equation:
38 = 2x + CE + 18
Now, let's solve for x and CE in terms of k and y.
First, substitute the value of x from the earlier equation:
38 = 2[(10 / 9) * (k * y)] + CE + 18
Simplifying:
38 = (20 / 9) * (k * y) + CE + 18
20 / 9 = (38 - 18 - CE) / (k * y)
(20 / 9) * (k * y) = 38 - 18 - CE
(20 / 9) * (k * y) = 20 - CE
Now, let's find the perimeter of FGHIJ using the scale factor:
Perimeter of FGHIJ = FG + GH + HI + IJ + JF = 9 + y + HI + 15 + 9
To find the length of HI, we can use the fact that the diagonal HJ has a length of 15:
Using Pythagorean Theorem:
(HI)^2 = (HJ)^2 - (IJ)^2
(HI)^2 = 15^2 - y^2
HI = sqrt(225 - y^2)
Therefore, the perimeter of FGHIJ is:
Perimeter of FGHIJ = 33 + sqrt(225 - y^2)
Finally, to find the length of CE:
(20 / 9) * (k * y) = 20 - CE
CE = 20 - (20 / 9) * (k * y)
I hope this helps you in assisting your granddaughter! Let me know if you need any further assistance.
To find the scale factor of pentagons ABCDE to FGHIJ, we need to compare corresponding side lengths.
Given that AE = 10 and FG = 9, we can start by finding the ratio of these side lengths. The scale factor k between AE and FG is given by: k = FG / AE.
k = 9 / 10 = 0.9
This means that the corresponding sides in the two pentagons are scaled by a factor of 0.9. Hence, for any side x in ABCDE, the corresponding side length in FGHIJ would be x * 0.9.
To find the value of x, we need to know the length of a side in ABCDE. You mentioned that the perimeter of ABCDE is 38, so we can divide the perimeter by 5 (since a pentagon has 5 equal sides) to find the length of a side:
Length of a side in ABCDE = Perimeter / 5 = 38 / 5 = 7.6
Therefore, x = 7.6 * 0.9 = 6.84
To find the perimeter of FGHIJ, we can use the same scale factor. Assuming GH is the corresponding side to AB, which is x, the length of GH would be x * 0.9.
Since the perimeter of ABCDE is 38, the perimeter of FGHIJ would be 38 * 0.9 = 34.2.
So the perimeter of FGHIJ is 34.2.
For the length of CE, we need to know the length of HJ, which you mentioned is 15. Assuming HJ is the corresponding side to CE, we know that HJ = CE * 0.9.
Therefore, CE = HJ / 0.9 = 15 / 0.9 = 16.67 (rounded to two decimal places).
So the length of CE is approximately 16.67.
To summarize:
- The scale factor of ABCDE to FGHIJ is 0.9.
- The value of x in ABCDE is approximately 6.84.
- The perimeter of FGHIJ is 34.2.
- The length of CE is approximately 16.67.