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?
+2668 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 \(\color{brown}\boxed{5\sqrt2 \over2}\)
+9674 I think the final answer is wrong, because:
\(\begin{array}{rcl} \text{length of diagonal} &=& \sqrt{\left(\dfrac{5 - 2\sqrt 3}2\right)^2 + \left(\dfrac{5 + 2\sqrt 3}2\right)^2}\\ &=& \dfrac12 \sqrt{(5 - 2\sqrt 3)^2 + (5 + 2\sqrt 3)^2}\\ &=& \dfrac12 \sqrt{(25 - 20\sqrt 3 + 12) + (25 + 20 \sqrt 3 + 12)}\\ &=& \dfrac{\sqrt{74}}2 \\&\neq& \dfrac{5\sqrt 2}2 \end{array}\)
+130071 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
