Why do most of the problems here contain 2018? (when it's still 2017)

Algebra Level 2

A right-angled triangle has side lengths of $a$ , $b$ and $2017$ (where $a < b < 2017$ ).

A square has a side length of $a + b$ , and its area is $4,072,323$ $( = 2017 \times 2019)$ . (Note the deliberate avoidance of the number 2018!!!)

What is the value of $a \times b$ ?

1 solution

Stephen Mellor
Dec 19, 2017

By Pythagoras' Theorem,

$a^2 + b^2 = 2017 \times 2017$

By the area of the square,

$(a + b)^2 = 2017 \times 2019$

$a^2 + b^2 + 2ab = (2017 \times 2017) + (2017 \times 2)$

Subtracting these two equations we get,

$2ab = 2017 \times 2$

$ab = \boxed{2017}$

(And the solution avoids typing big numbers into your calculator)