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Geometry

Properties of Triangles

Properties of Triangles: Level 3 Challenges

         

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

If \(y+4=z\) and \(z>x\) , then compute \(\dfrac{z}{x}\).

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

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

\(ABC\) is an isosceles triangle with \(AC = BC\). Furthermore, \(D\) is a point on \(BC\) that bisects the angle at \(A\). If \(\angle B = 72^\circ\) and \(CD=1,\) then find length of \(BD\)(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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