Rectangles ABCD and EFGH are similar. The perimeter of rectangle ABCD is 5 times greater than the perimeter of rectangle EFGH. What is the relationship between the areas of the rectangles?
Rectangle ABCD ~ rectangle EFGH, the perimeter of ABCD is 54 centimeters and the perimeter of EFGH is 36. what is the scale factor
similar figures with perimters in ratio 5 have areas in ratio 5^2
8o8u
To solve this problem, we need to understand the relationship between the perimeters and areas of similar rectangles. Two rectangles are similar if their corresponding sides are in proportion or have the same scaling factor.
Let's assume that the scaling factor between the sides of rectangles ABCD and EFGH is "x." This means each side of ABCD is x times greater than the corresponding side of EFGH.
If the perimeter of rectangle ABCD is 5 times greater than the perimeter of rectangle EFGH, the relationship between their perimeters can be expressed as:
Perimeter(ABCD) = 5 * Perimeter(EFGH)
2(AB+BC) = 5 * 2(EF+FG)
Since the sides of the rectangles are in proportion, we can say:
AB/EF = BC/FG = CD/GH = AD/EH = x
Now, let's find the relationship between their areas.
The area of a rectangle can be calculated by multiplying the length (l) and the width (w).
For rectangle ABCD:
Area(ABCD) = AB * BC
For rectangle EFGH:
Area(EFGH) = EF * FG
Since AB/EF = BC/FG = x, we can say:
AB = x * EF
BC = x * FG
Now, substituting these values into the area formulas:
Area(ABCD) = (x * EF) * (x * FG)
Area(ABCD) = x^2 * EF * FG
Area(EFGH) = EF * FG
Therefore, the relationship between the areas of the rectangles is:
Area(ABCD) / Area(EFGH) = (x^2 * EF * FG) / (EF * FG)
Area(ABCD) / Area(EFGH) = x^2
In conclusion, the relationship between the areas of rectangles ABCD and EFGH is that the area of ABCD is x^2 times greater than the area of EFGH, where x is the scaling factor between the sides of the rectangles.