The widths of two similar rectangles are 16 cm and 14 cm. What is the ratio of their perimeters? Of their areas?
Perimeter ratio: 8/7
area ratio=(8/7)^2
To find the ratio of the perimeters of two similar rectangles, you need to compare the sum of the lengths of their corresponding sides. Similarly, to find the ratio of their areas, you need to compare the product of the lengths and widths of the rectangles.
Let's denote the ratio of the perimeters as P, and the ratio of the areas as A.
First, let's find the corresponding side lengths of the two rectangles. Since the rectangles are similar, their corresponding sides are proportional.
Let the length of the first rectangle be L₁ and the width be W₁. Similarly, let the length of the second rectangle be L₂ and the width be W₂.
Given:
L₁ = ?
W₁ = 16 cm
L₂ = ?
W₂ = 14 cm
Since the rectangles are similar, we can set up the following proportion:
L₁ / W₁ = L₂ / W₂
Plugging in the given values, we have:
L₁ / 16 = L₂ / 14
To find L₁, we rearrange the equation:
L₁ = (16 * L₂) / 14
Now that we have the lengths of the corresponding sides, we can find P and A.
P (ratio of perimeters) = (L₁ + W₁) / (L₂ + W₂)
A (ratio of areas) = (L₁ * W₁) / (L₂ * W₂)
Plugging in the values we derived, we have:
P = [(16 * L₂) / 14 + 16] / [L₂ + 14]
A = [(16 * L₂) / 14 * 16] / [L₂ * 14]
Simplifying the expressions, we get:
P = (16L₂ + 224) / (14L₂ + 196)
A = (256L₂) / (14L₂)
The ratio of their perimeters (P) is equal to (16L₂ + 224) / (14L₂ + 196), and the ratio of their areas (A) is equal to (256L₂) / (14L₂).
To find the specific values of P and A, we need to know the length of either of the rectangles.