Correlation from a summed variance

EasyRandom Variables Statistics~3m

Two random variables have standard deviations $\sigma_X = 5$ and $\sigma_Y = 12$ . You are told that their sum has standard deviation $\sigma_{X+Y} = 17$ . What is the correlation $\rho(X,Y)$ ?

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Write $\operatorname{Var}(X+Y) = \sigma_X^2 + \sigma_Y^2 + 2\rho\,\sigma_X\sigma_Y$ and substitute the three variances.
When do standard deviations add exactly, $\sigma_{X+Y} = \sigma_X + \sigma_Y$ ? That case pins down $\rho$ on sight.

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Correlation from a summed variance

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