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# geometry

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A large square is divided into 4 small congruent rectangles and a small square as shown. The areas of the large and small squares are 25 and $$12$$, respectively. What is the length of a diagonal of a small rectangle? Apr 16, 2022

#1
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The big square has side length of 5.

The small square has a side length of $$\sqrt {12} = 2\sqrt3$$

The rectangles are congruent, so their width can be expressed as $$2x+2\sqrt3=5$$

Solving for x, we find $$x = {{5 - 2\sqrt3} \over 2}$$

The length can be expressed as $${2\sqrt3}+{5 -2\sqrt3 \over 2} = {5+2\sqrt3 \over2}$$

Using the Pythagorean Theorem, the length of the diagonal is $$\color{brown}\boxed{5\sqrt2 \over2}$$

Apr 16, 2022
#2
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I think the final answer is wrong, because:

$$\begin{array}{rcl} \text{length of diagonal} &=& \sqrt{\left(\dfrac{5 - 2\sqrt 3}2\right)^2 + \left(\dfrac{5 + 2\sqrt 3}2\right)^2}\\ &=& \dfrac12 \sqrt{(5 - 2\sqrt 3)^2 + (5 + 2\sqrt 3)^2}\\ &=& \dfrac12 \sqrt{(25 - 20\sqrt 3 + 12) + (25 + 20 \sqrt 3 + 12)}\\ &=& \dfrac{\sqrt{74}}2 \\&\neq& \dfrac{5\sqrt 2}2 \end{array}$$

Apr 16, 2022
#3
+1

Length of the side of  the  large square =   5

Length of the  side of  the  small square =  sqrt 12

Width of   rectangle =   (5  - sqrt 12)  / 2 =   (5/2)  - 2  sqrt (3) / 2  =  5/2 - sqrt 3

Length of rectangle   =  5  -  (5/2 - sqrt 3)  =  5/2 + sqrt (3)

Using the Pythagorean Theorem

Diagonal  =   sqrt [  ( 5/2  - sqrt 3)^2   + ( 5/2 + sqrt 3)^2  ]  =

sqrt  [  50/4  + 6  ]   =

sqrt  [  ( 50  + 24) / 4 ]  =

sqrt (74)  / 2   Apr 17, 2022