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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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06 Sep 2018, 00:09
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35% (medium)

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70% (01:19) correct 30% (01:21) wrong based on 31 sessions

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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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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)

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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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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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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)

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

$$P=\frac{4\sqrt{b}*\sqrt{2}}{\sqrt{2}*\sqrt{2}}=\frac{4\sqrt{2b}}{2}=2\sqrt{2b}$$

*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   [#permalink] 06 Sep 2018, 16:13
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