Trapezoid ABCD, top AD and bottom BC are parallel. AD is the smaller of the two parallel lines. Left AB is 45, bottom BC is 60, right CD is 30, top AD is unknown.
Trapezoid EFGH is similar to ABCD, top EH and bottom FG are parallel. EH is the smaller of the two parallel lines. Top EH is 10, other sides are unknown.
Area of ABCD is 9 times larger than EFGH.
What is the perimeter of EFGH ?
The areas of similar figures are proportional to the square of their corresponding sides
since the areas are in the ratio of 9:1
the sides are in the ratio of 3:1
so AD/EH = 3/1
AD/10 = 3/1
AD = 30
Perimeter of ABCD = 165
Perimeter EFGH/Perimeter ABCD = 1/3
Perimeter EFGH/165 = 1/3
perimeter EFGH = 55
I am still lost.
How come the areas of similar figures (trapezoids) are proportional to the squeare of their corresponding perimeters.
let me illustrate with an example
consider the right-angled triange with sides 3, 4, and 5
(easy to find areas of right-angles triangles)
a triangle similar is 12, 16, 20 ( I multiplied each side by 4)
area of smaller = (1/2)(3)(4) = 6
area of larger = (1/2)(12)(16) = 96
notice the ratio of their sides = 1:4
ratio of their areas = 6:96 = 1:16 = 1^2 : 4^2
which is the square of their sides.
now look at the perimeters
per of smaller = 3+4+5 = 12
per of larger = 12+16+20 = 48
notice the ratio of perimeters is 12:48 = 1:4
the same as the ratio of sides
Perimeter is a linear relationship
while area is a second degree relationship
Does that help?
Does this work for all trapezoids ?
Is this true for all similar enclosed figures ?
Can you help me with my other question posted minutes later than this one ?
To find the perimeter of trapezoid EFGH, we first need to find the lengths of the sides.
Let's start by finding the length of FH. Since trapezoids ABCD and EFGH are similar, they have proportional side lengths.
The ratio of the lengths of the corresponding sides of similar figures is equal to the scale factor between the figures. In this case, we know that the area of ABCD is 9 times larger than EFGH.
Since the area of a trapezoid is proportional to the product of the height and the average of the bases, we can set up the following equation:
(1/2)(AD + BC) = 9 * (1/2)(EH + FG)
We already know the lengths of AB, BC, and EH, so we can substitute those values into the equation:
(1/2)(45 + 60) = 9 * (1/2)(10 + FG)
(1/2)(105) = 9 * (1/2)(10 + FG)
105 = 9(10 + FG)
105 = 90 + 9FG
15 = 9FG
FG = 15/9
FG = 5/3
So, we have found the length of FG to be 5/3.
Next, we can find the length of GH. Since FG and EH are parallel and EH is the smaller of the two parallel lines, GH is also equal to 5/3.
Finally, to find the perimeter of trapezoid EFGH, we sum the lengths of all four sides:
Perimeter of EFGH = EH + FG + GH + EF
Perimeter of EFGH = 10 + 5/3 + 5/3 + EF
We still need to find the length of EF to calculate the perimeter. Could you provide the length of EF or any other known information about trapezoid EFGH?