We have a triangle which has sides of the lengths 15, 19 and 23. We make it into an obtuse triangle by shortening each of its sides by $x$. What is the range of values that $x$ can take? First, since $15 - x$, $19 - x$ and $23 - x$ can be the lengths of the sides of a triangle, it follows that $$x < \mathbf{AB}.$$ In addition, such a triangle is an obtuse triangle only when $x$ satisfies $$x^2 - \mathbf{CD}x + \mathbf{EF} < 0.$$ By solving this quadratic inequality, we have $$\mathbf{G} < x < \overline{\mathbf{HI}}.$$ Hence, the range of $x$ is $$\mathbf{J} < x < \mathbf{KL}.$$
We have a triangle which has sides of the lengths 15, 19 and 23. We make it into an obtuse triangle by shortening each of its sides by $x$. What is the range of values that $x$ can take?
First, since $15 - x$, $19 - x$ and $23 - x$ can be the lengths of the sides of a triangle, it follows that
$$x < \mathbf{AB}.$$
In addition, such a triangle is an obtuse triangle only when $x$ satisfies
$$x^2 - \mathbf{CD}x + \mathbf{EF} < 0.$$
By solving this quadratic inequality, we have
$$\mathbf{G} < x < \overline{\mathbf{HI}}.$$
Hence, the range of $x$ is
$$\mathbf{J} < x < \mathbf{KL}.$$