Answer:
Sides of both regular hexagon's are 2 cm , 3 cm.
Step-by-step explanation:
[tex]\boxed{\bf Area \ of \ regular \ hexagon = \dfrac{3\sqrt{3}a^2}{2}}[/tex]
a is the side of regular hexagon.
Let the side of one regular hexagon be 'x' cm.
The side of other regular hexagon = (x + 1) cm
[tex]\sf \text{Area of the first regular hexagon = $ \dfrac{3\sqrt{3}x^2}{2}$}\\\\\text{Area of the second regular hexagon = $ \dfrac{3\sqrt{3}(x+1)^2}{2}$}[/tex]
Product of their areas = 243
[tex]\sf ~~~~~~~~~ \dfrac{3\sqrt{3}x^2}{2}* \dfrac{3\sqrt{3}(x+1)^2}{2}=243\\\\\\ \dfrac{3*3* *\sqrt{3*3}*x^2 *(x + 1)^2 }{4}=243\\\\\\~~~~~~~~~~~~~ \dfrac{9*3*x^2*(x+1)^2}{4}=243\\\\\\~~~~~~~~~~~~~ \dfrac{27 *x^2*(x +1)^2}{4}=243\\\\\\~~~~~~~~~~~~~~~~~~~~ x^2 *(x +1)^2 = \dfrac{243*4}{27}\\\\\\~~~~~~~~~~~~~~~~~~~ x^2 *(x +1)^2 = 9*4\\[/tex]
x²* (x + 1)² = 36
Take square roots,
x *(x + 1) = 6
x² + x - 6 = 0
x² - 2x + 3x - 6 = 0
x(x - 2) + 3(x - 2) = 0
(x - 2)(x + 3) = 0
x - 2 = 0 ; x+ 3 = 0
x = 2 ; x = -3
x = -3 is ignored as measurement will not be in negative.
Side of one regular hexagon = 2 cm
Side of second regular hexagon = (2 +1 ) = 3 cm
