The area of square ABCD is x^ 2 + 18 √ 3 x + 243 . What is the length of the square's diagonal?
A) x+9√3
B) 4x+36√3
C) (x+9√3)^2
D)√2 (x+9√3)
E) (x+9√3)/√2
Is there a strategic approach to this question? Can any experts show?
Hi ardz24,
Let's take a look at your question.
We know that the area of a square is always the square of the length of side.
So we can find the length of side using the area.
$$Area=x^2+18\sqrt{3}x+243$$
If s represents the length of side of square, then,
$$s^2=x^2+18\sqrt{3}x+243$$
Use the formula a^2+2ab+b^2=(a+b)^2 to write the RHS as a square:
$$s^2=x^2+2\left(x\right)\left(9\sqrt{3}\right)+\left(9\sqrt{3}\right)^2$$
$$s^2=\left(x+9\sqrt{3}\right)^2$$
$$s=\left(x+9\sqrt{3}\right)$$
Now we can find the length of the diagonal of the square using Pythagorean theorem,
$$\left(Diagonal\right)^2=\left(x+9\sqrt{3}\right)^2+\left(x+9\sqrt{3}\right)^2$$
$$Diagonal=\sqrt{2\left(x+9\sqrt{3}\right)^2}$$
$$Diagonal=\sqrt{2}\left(x+9\sqrt{3}\right)$$
Therefore, option
D is correct.
Hope it helps.
I am available if you'd like any follow up.
