Question

In the diagram, RSTU~ ABCD. Find the ratio of their perimeters.

U
R
18 S
36
T
D
A
24
The ratio of their perimeters is
B
C

Answer 1

Answer: Let's call the ratio of the perimeters of the two similar figures "k". That means that for every unit of length in RSTU, there are k units of length in ABCD.

Since UR = 18 and DA = 24, k = 24/18 = 4/3.

So for every length in RSTU, there are 4/3 times that length in ABCD. We can use this ratio to find the corresponding lengths in ABCD. For example, SR = 36, so BS = 36 * (4/3) = 48.

Now that we have the lengths of all sides of ABCD, we can find its perimeter:

Perimeter of ABCD = 24 + 48 + BC + 24 = 120 + BC

Perimeter of RSTU = 18 + 36 + 18 + 36 = 108

So the ratio of their perimeters is:

Perimeter of ABCD / Perimeter of RSTU = (120 + BC) / 108 = (120 / 108) + (BC / 108) = 20/18 + BC/108

Since the perimeters of the two figures are similar, the ratio of their perimeters is equal to the ratio of their corresponding side lengths:

20/18 + BC/108 = 4/3

Solving for BC:

BC = (4/3 - 20/18) * 108 = 36

Explanation:

Answer 2

The ratio of the perimeters of RSTU to ABCD is 3:2.

In the context of geometry, when two figures are similar, it means that they have the same shape but may differ in size. The given diagram indicates that the quadrilateral RSTU is similar to the quadrilateral ABCD. The corresponding sides of these two quadrilaterals are proportional. To find the ratio of their perimeters, we can sum up the lengths of the corresponding sides for each quadrilateral.

In this case, the ratio of the perimeters is determined by adding the lengths of RS, ST, TU, and UR for RSTU, and AB, BC, CD, and DA for ABCD. The resulting ratio of these perimeters is 3:2, indicating that the perimeters of RSTU and ABCD are in proportion with a factor of 3 to 2, reflecting the similarity of the two quadrilaterals.