Geometry

Properties of Triangles

Properties of Triangles: Level 3 Challenges

         

In a certain right triangle, the hypotenuse equals (x+z)(x+z) while the other two sides are (x+y)(x+y) and (y+z)(y+z) for positive integers x,y,zx, y, z.

If y+4=zy+4=z and z>xz>x , then compute zx\dfrac{z}{x}.

Consider a rectangle which has a diagonal of length 6. If PP is a point on side ABAB such that AP=AD,\lvert\overline{AP}\rvert=\lvert\overline{AD}\rvert, and Q Q is a point on the extension of side ADAD such that AQ=AB,\lvert\overline{AQ}\rvert=\lvert\overline{AB}\rvert, what is the area of the quadrilateral APCQ?APCQ?

In triangle ABC,ABC, AB=ACAB = AC and BAC=100.\angle BAC = 100^\circ. If AB\overline{AB} is extended to DD such that AD=BC,AD=BC, find BCD\angle BCD (in degrees).

ABCABC is an isosceles triangle with AC=BCAC = BC. Furthermore, DD is a point on BCBC that bisects the angle at AA.

If B=72\angle B = 72^\circ and CD=1,CD=1, then find length of BDBD (upto 3 decimal places).

In a triangle with integer side lengths, one side is three times as long as a second side, and the length of the third side is 17. What is the greatest possible perimeter of the triangle?

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