Explanation:
First one
say the dimensions are x * h
for the shaded pentagon x * h = 4590
for the unshaded one d * x * d *h = 510 ( 'd' is the dilation factor)
d^2 * x * h = 510
but you know x * h = 4590 (underlined above)
so d^2 * 4590 = 510 <=====solve for 'd'
d^2 = 510/4590
d^2 = 1/9
d = 1/3
So each dimension is scaled by 1/3
21 m in the first pentagon is scaled by 1/3 to x in the second pentagon
1/3 * 21 = x = 7 m
Think you can do the triangle one now? (It's a little different)
area = 1/2 x * y = 16
new area = 1/2 d * x * d * y = d^2 * 1/2 x*y
d^2 16 = 71
d = 2.1065
so multiply the dimensions of triangle 1 by 2.1065to get the new larger triangle 2.1065 * x = 9.5 shows x = 4.5 in