Smartphone screen
Can you find the length and width of the screen of this smartphone in inches?
Problem
A smartphone's display is given as the diagonal distance between the vertices. A certain smartphone has a 5 inch display and a resolution of 1920 pixels (length) x 1200 pixels (width).
What are the length and width of the display, in inches? Leave your answer in the form $\dfrac{a}{\sqrt b}$, where $a$, $b$ are integers.
This problem is adapted from the World Mathematics Championships
Student Solutions
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.
Using a unit conversion between inches and pixels
Pythagoras' theorem can be used to find the the display of the smartphone in pixels.
$240\sqrt{89}$ can be obtained without a calculator using factorisation:
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
And $q=\dfrac{5}{8}p=\dfrac{5}{8}\dfrac{40}{\sqrt{89}}=\dfrac{25}{\sqrt{89}}$
Image

Using a unit conversion between inches and pixels
Pythagoras' theorem can be used to find the the display of the smartphone in pixels.
$240\sqrt{89}$ can be obtained without a calculator using factorisation:
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
And $q=\dfrac{5}{8}p=\dfrac{5}{8}\dfrac{40}{\sqrt{89}}=\dfrac{25}{\sqrt{89}}$