Smartphone screen

It is helpful to begin with a diagram of the smartphone. The red lengths are given in pixels, and the green length is given in inches.

 

Image

 

Using a unit conversion between inches and pixels

Pythagoras' theorem can be used to find the the display of the smartphone in pixels.

$$\begin{array}{rl}& {1920}^2+{1200}^2=c^2\\ \Rightarrow & 5126400=c^2\\ \Rightarrow & c=\sqrt{5216400}=240\sqrt{89}\approx 2264.2\end{array}$$

$240\sqrt{89}$ can be obtained without a calculator using factorisation:

$$\begin{array}{rl}& {1920}^2+{1200}^2=c^2\\ \Rightarrow & (20\times 96{)}^2+(20\times 60{)}^2=c^2\\ \Rightarrow & {20}^2\times {96}^2+{20}^2\times {60}^2=c^2\\ \Rightarrow & {20}^2\times ({96}^2+{60}^2)=c^2\\ \Rightarrow & {20}^2\times 4^2\times ({24}^2+{15}^2)=c^2\\ \Rightarrow & {20}^2\times 4^2\times 3^2\times (8^2+5^2)=c^2\\ \Rightarrow & (20\times 4\times 3{)}^2\times (64+25)=c^2\\ \Rightarrow & {240}^2\times 89=c^2\end{array}$$

So $c=240\sqrt{89}$ pixels, which is $5$ inches, and so each pixel must be $\dfrac{5}{240\sqrt{89}}$ inches.

So $1920$ pixels must be $\dfrac{1920\times5}{240\sqrt{89}}=\dfrac{40}{\sqrt{89}}$ inches,

and $1200$ pixels must be $\dfrac{1920\times5}{240\sqrt{89}}=\dfrac{25}{\sqrt{89}}$ inches.

Using the ratio between the sides

Calling the length and width in inches $p$ and $q$, we know that $\dfrac{q}{p}=\dfrac{1200}{1920}$, which simplifies to $\dfrac{5}{8}$. So if $\dfrac{q}{p}=\dfrac{5}{8}$, then $q=\dfrac{5}{8}p$.

By Pythagoras' theorem, $p^2+q^2=5^2$, so $$\begin{array}{rl}& p^2+{\left(\frac{5}{8}p\right)}^2=5^2\\ \Rightarrow & p^2+\frac{25}{64}{p}^2=5^2\\ \Rightarrow & \frac{64p^2+25p^2}{64}=25\\ \Rightarrow & \frac{89p^2}{64}=25\\ \Rightarrow & 89p^2=25\times 64\\ \Rightarrow & p^2={\displaystyle \frac{5^2\times 8^2}{89}}\\ \Rightarrow & p=\sqrt{{\displaystyle \frac{5^2\times 8^2}{89}}}={\displaystyle \frac{5\times 8}{\sqrt{89}}}={\displaystyle \frac{40}{\sqrt{89}}}\end{array}$$

And $q=\dfrac{5}{8}p=\dfrac{5}{8}\dfrac{40}{\sqrt{89}}=\dfrac{25}{\sqrt{89}}$