Final answer:
The perimeter of similar quadrilateral EFGH is calculated by determining the scale factor from the given side ratios, applying it to each side of ABCD, and then summing up the corresponding sides' lengths in EFGH to get a perimeter of 96.
Step-by-step explanation:
We are given two similar quadrilaterals, ABCD and EFGH, with the side lengths of quadrilateral ABCD being 2, 5, 7, and 10. The ratio of corresponding sides CD:GH is 1:4. To find the perimeter of EFGH, we first need to establish the scale factor based on the given side ratio.
The length of CD in quadrilateral ABCD is 7. Using the side ratio, we calculate the length of GH in quadrilateral EFGH as 7\(\times\)4 = 28. Now, since both quadrilaterals are similar, all their corresponding sides are in proportion, meaning we can use the same scale factor (4 in this case) to find the lengths of all sides of EFGH.
- EF = AB\(\times\)4 = 2\(\times\)4 = 8
- FG = BC\(\times\)4 = 5\(\times\)4 = 20
- GH is already calculated as 28
- HE = AD\(\times\)4 = 10\(\times\)4 = 40
Finally, to find the perimeter of quadrilateral EFGH, we add up all its sides:
Perimeter of EFGH = EF + FG + GH + HE = 8 + 20 + 28 + 40 = 96
Therefore, the perimeter of quadrilateral EFGH is 96.