Trapezoid ABCD is dilated to create trapezoid
by a scale factor of 1/2
.
Find the perimeter and area of both figures.
Dilation of trapezoid ABCD into trapezoid EFGH
Calculate the area of trapezoid EFGH
by using the square of k
.
To find the perimeter of a trapezoid, we need to add up the lengths of all its sides.
Let's assume that the lengths of the sides of trapezoid ABCD are AB, BC, CD, and DA, and the lengths of the sides of trapezoid EFGH are EF, FG, GH, and HE.
The scale factor of 1/2 means that the lengths of the corresponding sides in trapezoid EFGH will be half the lengths of the corresponding sides in trapezoid ABCD.
Therefore, we have:
EF = AB/2
FG = BC/2
GH = CD/2
HE = DA/2
To find the perimeter of trapezoid ABCD, we add up the lengths of its sides:
Perimeter of ABCD = AB + BC + CD + DA
We can substitute the values of EF, FG, GH, and HE into the equation to find the perimeter of trapezoid EFGH:
Perimeter of EFGH = EF + FG + GH + HE
= AB/2 + BC/2 + CD/2 + DA/2
= (AB + BC + CD + DA)/2
= (Perimeter of ABCD)/2
So, the perimeter of trapezoid EFGH is half the perimeter of trapezoid ABCD.
To find the area of a trapezoid, we need to know its height and the lengths of its parallel bases.
If h is the height of trapezoid ABCD, and the scale factor is k = 1/2, then the height of trapezoid EFGH will be h/2.
The lengths of the parallel bases in trapezoid ABCD are AB and CD, and in trapezoid EFGH, the lengths of the parallel bases will be AB/2 and CD/2.
To find the area of trapezoid ABCD, we use the formula:
Area of ABCD = (AB + CD) * h /2
Similarly, we can calculate the area of trapezoid EFGH using its lengths of parallel bases and the height:
Area of EFGH = (AB/2 + CD/2) * h/2
= (AB + CD) / 4
= (Area of ABCD) / 4
Therefore, the area of trapezoid EFGH is one-fourth the area of trapezoid ABCD.
To calculate the area of trapezoid EFGH using the square of k, we can use the formula:
Area of EFGH = (k^2) * Area of ABCD
= (1/2)^2 * Area of ABCD
= 1/4 * Area of ABCD
So, the area of trapezoid EFGH using the square of k is one-fourth the area of trapezoid ABCD.