Question

The figures below are similar. What are a) the ratio of the perimeters and b) the ratios of the areas of the larger figure to the smaller figure? The figures are not drawn to scale.
http://gyazo.com/53b9d9f3f176ae4b195f5cd8c431505f

Answer 1

ratio pperimeter is the same as the ratio of the corresponding sides of the larger and the smaller figure
whereas the ratio of the areas is the square of the ratio of the corresponding sides of the larger and the smaller figure

Answer 2

..I don't understand, to be honest.

Answer 3

Okay, two parts, perimeter, and area.

Perimeter: when we scale the edge length (multiply it by some number), we scale the edge length of each edge. So, if one edge goes from x to 2x, each of the edges of the figure will be multiplied by 2. As the perimeter is the sum of all of the edge lengths, by the distributive property we can see that the perimeter is also multiplied by 2.

Say we had a triangle with sides 3, 4, 5. The starting perimeter is \(P = 3+4+5 = 12\).

Now we multiply each side by 2, giving us sides 6, 8, 10. The new perimeter is \(P_{new} = 6+8+10 = 24\) which is easily seen to be twice the old perimeter, either by comparing the perimeters, or observing that

\[P_{new} = 2*3+2*4+2*5 = 2(3+4+5) = 2*P = 2*12=24 \]

Answer 4

Now for the area, which is a bit trickier. The area of the figure has two dimensions, length and width. When we double the length of each side, we are going to double both our length and our width. Because the area of a rectangle is given by \[A = l*w\]when we double the dimensions of our 2-dimensional figure (and thus double the size of the rectangle that fits around it), we are doubling the two items that go into computing the area. Thus we get
\[A_{new} = (2l)*(2w) = 4*l*w = 4*(l*w) = 4*A\]

Our area is 4 times the area of the original figure, or a ratio of 4:1.