Describe the effect of each change on the area of the given figure.

Question

If the diagonals of a Rhombus are both multiplied by 8.

If the base of a rectangle is multiplied by 4, and the height is multiplied by 7.

If the perimeter of an equilateral triangle is doubled.

Answer 1

For all of these questions, I would suggest using example numbers. You can, of course, use algebra, but I think this is the easiest way. For example:

1) The area of a rhombus = (D1 * D2) / 2. Say D1 = 3, D2 = 2. In this case, A = (3*2) / 2 = 3.

Multiplying the diagonals by 8, we see: D1 = 24, D2 = 16.

A = (24 * 16) / 2 = 192.

A1 / A2 = 192 / 3 = 64.

So, by multiplying both diagonals by 8, we see a 64 times increase in the area. This make sense considering we're changing the area by two multiples of 8, which equals 64.

2) Let's do the same thing with a rectangle. Say the base = 2 and the height = 5: A = 2 * 5 = 10

If we add the conditions, b = 2 * 4 = 8; h = 5 * 7 = 35.

A = 8 * 35 = 280.

A2 / A1 = 28.

So, this increases by a multiple of 28. Again, because we're changing the area by 4 times the base and 7 times the height, it should make sense that the area increases by 28 times (4 * 7 = 28).

3) Say each side of the equilateral triangle is 4, the P = 12.

But this is asking for the area, so, using the area of an equilateral triangle formula, which is (√3 / 4) * s².

So, A = (√3 / 4) * 4^2. = (√3 / 4) * 16 = 4√3

If the perimeter is doubled, then P = 24. And s would have to equal 8.

So! A = (√3 / 4) * 8^{2 =} (√3 / 4) * 64 = 16√3

Thus, this area increases by a multiple of 4.