Question

a quilt design has one large rectangle surrounded by four congruent rectangles, similar to the large rectangle. If the large rectangle has an area of 45 square inches, what is the area of each small rectangle?

Answer 1

Area of each smaller rectangle = 5inches^2

Explanation:

Bigger rectangle:

Area (A)= 45inches^2

Length (L)= 9inches

Breadth (B) = area/length = 45/9

Breadth = 5inches

Smaller rectangle:

Area (a)= unknown

Length (l) = 3inches

Breadth (b)= unknown

Since the bigger rectangle is similar to the 4 smaller congruent rectangles, we find ratio of the length and breath of smaller rectangle to the bigger one.

Scale factor is calculated from any two sides (length or breadth).

Ratio of length = l/L = 3/9 = 1/3

Ratio of breadth = b/B = b/5

Sum of polygon theorem states that if the ratio of two figures is c/d, the ratio of the area of the two similar polygons is equal to the square of the ratio of the two figures (c/d)^2.

a/A = (l/L)^2 = (b/B)^2

l/L = b/B (similar polygons)

1/3 = b/5

3b = 5

b = 5/3

Smaller breadth = 5/3

Area of smaller rectangle (a) = l × b

Area of smaller rectangle = 3 × 5/3

Area of each smaller rectangle = 5inches^2

Answer 2

Area of each small rectangle = 5 in²

Explanation:

Let the length of the large rectangle be L

Let the breadth of the large rectangle be B

The small rectangles are said to be congruent, that is, they have the same shape and sizes.

Let the length of the small rectangles each be l

And their breadths, each be b

Area of the large rectangle = 45 in²

Side length of the large rectangle = 9 in.

Area of a rectangle = (length) × (breadth)

A = L × B

45 = 9 × breadth

Breadth = B = (45/9) = 5 in.

The small rectangles are similar to the large rectangle, that is, the ratio of their sizes are the same.

So,

(l/b) = (L/B)

or

(l/L) = (b/B)

l = 3 in. (Given in the diagram of the question)

L = 9 in.

b = ?

B = 5 in.

(3/b) = (9/5)

b = (3×5/9) = (5/3) in.

Area of each of the small rectangles = l × b

Area of each small rectangle = 3 × (5/3)

= 5 in²

Hope this Helps!!!