Classification of Triangles

Classification of Triangles: Level 3 Challenges


For some value d,d, an isosceles triangle has at least one angle of (2d40)(2d-40) degrees and at least one angle of (d+20)(d+20) degrees.

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

The sides of a triangle have lengths 11,15,k11 , 15 , k where kk is an integer. For how many value(s) of kk is the triangle obtuse?

This question is a part of JEE Novices.

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

Find the largest angle of the triangle in degrees.

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

The two larger angles of an isosceles triangle have a combined measure of 135135^\circ. If this triangle has a perpendicular height of 2 m2\text{ m}, find the length of its base in meters.

If the latter can be expressed in the form abca\sqrt { b } -c, where a,ba,b and cc are positive integers with bb square-free, write your answer as a×b×ca\times b\times c.


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