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Can a scalene triangle be obtuse? Is an equilateral triangle always acute?

For some value \(d,\) an isosceles triangle has at least one angle of \((2d-40)\) degrees and at least one angle of \((d+20)\) degrees.

For each of such triangles, let \(\alpha\) denote the smallest angle in degrees. Then what is the sum of all possible \(\alpha\)'s?

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The sides of a triangle have lengths \(11 , 15 , k\) where \(k\) is an integer. For how many value(s) of \(k\) is the triangle obtuse?

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A triangle has sides of length \(\sin( x), \cos (x)\) and \(\sqrt{1+\sin (x )\cos( x)}\), with \(x\) an acute angle.

Find the largest angle of the triangle in degrees.

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A line \(AB\) of length \(l\) makes up one side of a triangle. The locus of all points \(P\) such that \(\triangle ABP\) is a right triangle with \(\angle P = 90^{\circ}\) encloses an area of \( \text{__________} \).

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The two larger angles of an isosceles triangle have a combined measure of \(135^\circ\). If this triangle has a perpendicular height of \(2\text{ m}\), find the length of its base in meters.

If the latter can be expressed in the form \(a\sqrt { b } -c\), where \(a,b\) and \(c\) are positive integers with \(b\) square-free, write your answer as \(a\times b\times c\).

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