1. A rhombus has an area of 9cm2. The area is multiplied by 5. Describe the effects on the diagonals of the rhombus.
2. A circle has a circumference of 14pi ft. The area is halved. Describe the effects on the circumference.
1. The area of a rhombus - in terms of its diagonals - is :
[d1* d2] / 2 where d1 and d2 are the diagonal lengths
So.......the original area has been quintupled.......the effect on each diagonal is to increase the original length by √5......to see why this is so......let the original area =
[d1 * d2]/ 2
Now....increase each diagonal lenght by √5 and we have
Area = [√5d1 * √5d2] / 2 = 5 [d1*d2] / 2 = 5 times the original area
2. If the area is halved.......the circumference is 1/√2 as much as before
To see this.......note that if the original circumference was 14pi ft....the radius = 7 ft
And the original area = pi*r^2 = pi*49 = 49pi ft^2
So.....1/2 of this = 49pi/2 ft^2.....and we can solve for the new radius thusly :
49pi/2 = pi *r^2 divide by pi on both sides
r^2 = 49/[2]
r = 7/ [√2] ft
So....the new circumference = 2 * pi * [7/√2] = 14pi/√2 = 14pi * (1/√2) ft