Trapezoid ABCD is dilated to create trapezoid EFGH by a scale factor of 1/2,
.
Find the perimeter and area of both figures.
Dilation of trapezoid ABCD into trapezoid EFGH
Find the dimensions of trapezoid EFGH
.
Side EF corresponds to side AB.mEF=k(mAB)+mEF=?
.
Given that the scale factor of the dilation is 1/2, we can determine the lengths of the corresponding sides of trapezoid EFGH by multiplying the lengths of the corresponding sides of trapezoid ABCD by 1/2.
Let's assume that side AB has a length of x.
Side EF corresponds to side AB, so its length can be found by multiplying the length of AB by 1/2:
mEF = (1/2)(mAB) = (1/2)(x) = x/2
Therefore, side EF has a length of x/2.
Similarly, side HG corresponds to side CD, and its length can be found by multiplying the length of CD by 1/2:
mHG = (1/2)(mCD) = (1/2)(x) = x/2
Therefore, side HG also has a length of x/2.
Side FG corresponds to side BC, and its length can be found by multiplying the length of BC by 1/2:
mFG = (1/2)(mBC) = (1/2)(x) = x/2
Therefore, side FG has a length of x/2.
Side EH corresponds to side AD, and its length can be found by multiplying the length of AD by 1/2:
mEH = (1/2)(mAD) = (1/2)(x) = x/2
Therefore, side EH also has a length of x/2.
To determine the perimeter of trapezoid EFGH, we need to sum up the lengths of all its sides:
Perimeter of EFGH = EF + FG + GH + EH
= (x/2) + (x/2) + (x/2) + (x/2)
= 2x/2 + 2x/2
= (2x + 2x)/2
= 4x/2
= 2x
Therefore, the perimeter of trapezoid EFGH is 2x.
To determine the area of trapezoid EFGH, we need to use the formula for the area of a trapezoid:
Area of EFGH = (1/2)(sum of the lengths of the parallel sides)(height)
= (1/2)(EF + GH)(height)
Given that trapezoid EFGH is a dilation of trapezoid ABCD with a scale factor of 1/2, the height of trapezoid EFGH must also be half of the height of trapezoid ABCD.
So, if the height of trapezoid ABCD is h, then the height of trapezoid EFGH is (1/2)(h) = h/2.
Using the previously determined lengths of the corresponding sides, we can write the formula for the area of trapezoid EFGH as:
Area of EFGH = (1/2)(EF + GH)(h/2)
= (1/2)((x/2) + (x/2))(h/2)
= (1/2)(x)(h/2)
= (1/4)(xh)
Therefore, the area of trapezoid EFGH is (1/4)(xh).