Squares in Rectangle

Geometry Level 3

Rectangle $ABCD$ is divided into squares.The length of side $AB= 16$ cm. What is length of side $AD$ (in cm) ?

\dfrac {29}{2}

13

14

\dfrac {27}{2}

15

1 solution

Steven Chase
Dec 3, 2018

By inspection, we can form the following equations. I would have liked to have six independent linear equations, but the sixth one didn't jump out to me, so it is non-linear.

$a + 2 b = 16 \\ d + c = 16 \\ d + e + a = b + c \\ 2 e + f = d \\ a + e - f = b \\ (d+ c)(c + b) = a^2 + 2 b^2 + c^2 + d^2 + 2 e^2 + f^2$

There are multiple solutions to these equations, but $b + c$ is equal to $\frac{29}{2}$ for solutions with the right relative sizes. For example, $a$ must be less than $b$ .

Addendum:

Thanks to @Jordan Cahn for pointing out that the sixth independent linear equation is the following:

$2 b = c + f$

The overall system is therefore:

$a + 2 b = 16 \\ d + c = 16 \\ d + e + a = b + c \\ 2 e + f = d \\ a + e - f = b \\ 2 b = c + f$

I don't agree that $b+c=\frac{29}{2}$ for every solution. $(a,b,c,d,e,f) = \left(\frac{16}{3}, \frac{16}{3}, \frac{64}{7}, \frac{48}{7}, \frac{16}{7}, \frac{16}{7}\right)$ is a solution to your system, but has $b+c = \frac{304}{21}$ .

The sixth independent linear equation you're looking for is $2b=c+f$ . That gives the system a unique solution with $b+c=\frac{29}{2}$ .

Jordan Cahn - 2 years, 6 months ago

Good catch, thanks. My solver finds that solution too. I have qualified my previous statement with a note about screening solutions based on size relationships, and I have added an addendum mentioning the sixth linear equation.

Steven Chase - 2 years, 6 months ago

Thanks for the solution :)

A Former Brilliant Member - 2 years, 6 months ago

Squares in Rectangle

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