Help !!! I have trouble doing this?

Question

PQRS is a rhombus.

The length of PR is (3 √3 + 2 √2) cm and the length of QS is (3 √3 - 2 √2) cm.

(i) Find the area of the rhombus (What's the formula to find area of rhombus?)

(ii) Express the perimeter of the rhombus in the form of a √b cm where a and b are integers.


Please show your workings clearly..thank you

Answer 1

We are talking about a rhombus. You can see it made up of two identical triangles, each of which has area computed as
PR*(1/2*QS)*1/2=1/4*PR*QS. And the area of the rhombus is twice that big, that is, 1/2*PR*QS. In the problem,

1/2*PR*QS
=1/2(3 √3 + 2 √2)((3 √3 - 2 √2))
=1/2*(27-8)
=19/2.

A rhombus has four identical sides. The length of each side is computed as sqrt[(1/2*PR)^2+(1/2*QS)^2]. This can be understood by considering a rhombus made up by four identical small right-angle triangles. For each, you know two of the sides (a, b, for instance), then the longest side c (also the side of the rhombus) should be c=sqrt(a^2+b^2).

Hence, each side is equal to

sqrt[1/4*(27+8+12sqrt(6)
+27+8-12sqrt(6)]
=sqrt[1/4*70]
=1/2*sqrt(70)

The perimeter is 4 times the length of each side and is therefore 2*sqrt(70).

Answer 2

A =1/2 D*d (D and d diagonals)
A= 27-8 =19cm^2 Remember (a+b)*(a-b) = a^2-b^2)
The side is
1/2 sqrt ( 2*(27+8)) = 1/2 sqrt(70) (Pythagoras) so
Perimeter = 2sqrt(70) cm