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Properties of Triangles

Given a triangle with side lengths of 5, 12, and 14, is the largest angle in the triangle acute, right, or obtuse? Geometric knowledge helps us deduce much about triangles from limited information.

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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