Can someone me ? 1. by what factor does the area change if both diagonals are doubled? explain.

The area of the figure is half the product of the diagonals:
  A = (1/2)*(5 cm)*(10 cm) = 25 cm²

If both diagonals are doubled, each of the dimensions in the above equation is multiplied by 2, so two factors of 2 are thrown into the product. The area of the larger figure will be 2*2 = 4 times that of the figure shown:
  A = (1/2)*(10 cm)*(20 cm) = 100 cm²
  100 cm² = 4*(25 cm²)

If both diagonals are doubled, the area changes by a factor of 4.


When every linear dimension of a geometric figure is multiplied by the same scale factor (2, in this case), the new figure is "similar" to the old one. Any area measure of the new figure will be the square of that scale factor times the original area. Any volume of the new figure will be the cube of that scale factor times the original volume.