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If a square has a diagonal of length b, what is the perimeter of the

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If a square has a diagonal of length b, what is the perimeter of the  [#permalink]

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New post 06 Sep 2018, 00:09
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If a square has a diagonal of length \(\sqrt{b}\), what is the perimeter of the square?


A. 4b

B. \(4\sqrt{2b}\)

C. \(2\sqrt{\frac{b}{2}}\)

D. \(\frac{4\sqrt{b}}{b}\)

E. \(2\sqrt{2b}\)

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If a square has a diagonal of length b, what is the perimeter of the  [#permalink]

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New post 06 Sep 2018, 02:02
Bunuel wrote:
If a square has a diagonal of length \(\sqrt{b}\), what is the perimeter of the square?


A. 4b

B. \(4\sqrt{2b}\)

C. \(2\sqrt{\frac{b}{2}}\)

D. \(\frac{4\sqrt{b}}{b}\)

E. \(2\sqrt{2b}\)


For a square with side a, diagonal = a√2

i.e. a√2 = √b
i.e. a = √(b/2)

Perimeter of square = 4a = 4*√(b/2) = 2*2*√(b/2) = 2*√(4b/2) = 2√(2b)

Answer: Option E
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Re: If a square has a diagonal of length b, what is the perimeter of the  [#permalink]

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New post 06 Sep 2018, 02:51
Bunuel wrote:
If a square has a diagonal of length \(\sqrt{b}\), what is the perimeter of the square?

+1 for E.

Area of Square = Diagonal^2 / 2 = √b^2 / 2 = b/2
Area of Square = S^2 = b/2 --> S = √b / √2
Perimeter of Square = 4 * S = (4 * √b) / √2 = ((√2*√2*√2*√2) * √b) / √2 = √2*√2*√2 * √b = 2* √2 * √b = 2√2b

Hence, E.
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If a square has a diagonal of length b, what is the perimeter of the  [#permalink]

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New post 06 Sep 2018, 16:13
Bunuel wrote:
If a square has a diagonal of length \(\sqrt{b}\), what is the perimeter of the square?


A. 4b

B. \(4\sqrt{2b}\)

C. \(2\sqrt{\frac{b}{2}}\)

D. \(\frac{4\sqrt{b}}{b}\)

E. \(2\sqrt{2b}\)

Perimeter of a square = \(4s\)

Side of the square is derived from the diagonal*. This square: diagonal, \(d=\sqrt{b}\)
The diagonal of any square
\(d=s\sqrt{2}\) so
\(\sqrt{b}=s\sqrt{2}\)
\(\frac{\sqrt{b}}{\sqrt{2}}=s\)

Perimeter, \(P=4s=(4*\frac{\sqrt{b}}{\sqrt{2}})=\frac{4\sqrt{b}}{\sqrt{2}}\)
No answers match. Rationalize the denominator (get rid of the radical sign)

\(P=(\frac{4\sqrt{b}}{\sqrt{2}})*(\frac{\sqrt{2}}{\sqrt{2}})\)

\(P=\frac{4\sqrt{b}*\sqrt{2}}{\sqrt{2}*\sqrt{2}}=\frac{4\sqrt{2b}}{2}=2\sqrt{2b}\)

Answer E

*If you do not know a square's side/diagonal relationship, use the Pythagorean theorem:
\(s^2+s^2=d^2\)
\(2s^2=d^2\)
\(\sqrt{2*s^2}=\sqrt{d^2}\)
\(s\sqrt{2}=d\)

OR use 45-45-90 triangle properties. Sides opposite those angles are in the ratio of \(s:s:s\sqrt{2}\).
The diagonal, opposite the 90° angle, corresponds with \(s\sqrt{2}\)
\(s\sqrt{2}=\sqrt{b}\)
\(s=\frac{\sqrt{b}}{\sqrt{2}}\)
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If a square has a diagonal of length b, what is the perimeter of the &nbs [#permalink] 06 Sep 2018, 16:13
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