CAT 2021 Slot 2 · Quant · Question 9

The diagonals of a rhombus bisect each other at right angles, so the half-diagonals and a side form a right triangle. Let the diagonals be {d}_{1} and {d}_{2}.

Area gives \frac{{d}_{1}{d}_{2}}{2}=12, so {d}_{1}{d}_{2}=24.

The right triangle gives {\left( \frac{{d}_{1}}{2}\right)}^{2}+{\left( \frac{{d}_{2}}{2}\right)}^{2}={5}^{2}, so {d}_{1}^{2}+{d}_{2}^{2}=100.

Now combine them rather than solving a quartic: {\left({d}_{1}+{d}_{2}\right)}^{2}=100+48=148,

so {d}_{1}+{d}_{2}=2\sqrt{37}, and {\left({d}_{1}-{d}_{2}\right)}^{2}=100-48=52,

so {d}_{1}-{d}_{2}=2\sqrt{13}.

Adding, 2{d}_{1}=2\sqrt{37}+2\sqrt{13},

so the longer diagonal is {d}_{1}=\sqrt{37}+\sqrt{13}.