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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.

Triangles Problem Solving


As shown in the above diagram, Raj folds a rectangular paper \(ABCD\) in such a way that side \(BC\) becomes \(BE.\) If the length of \(\overline{AB}\) is \(3\) and the length of \(\overline{BC}\) is \(15,\) what is the area of \(\triangle ABF?\)

Note: The above diagram is not drawn to scale.

In the above diagram, we are given \[\angle BAC=90^\circ, \angle ACD=75^\circ, \angle CBD=75^\circ, \angle BDE=110^\circ.\] What is \(\angle ABC\) in degrees?

In right triangle \(ABC\), we are given that \(\angle ABC = 90^\circ\) and \(AC= 34\). \(D\) is a point on line segment \(BC\) such that \(BD=12, DC=18\). What is the length of \(AD\)?

In the above diagram, the ratio of the length of \(\overline{AD}\) to the length of \(\overline{DC}\) is \(3:2.\) If the area of \(\triangle ABE\) is \(13,\) what is the area of \(\triangle BEC ?\)

Note: The above diagram is not drawn to scale.

In triangle \(ABC\), we have \(AB = 47, BC = 48, CA = 49 \). Points \(D, E,\) and \(F\) are the midpoints of \(BC, CA,\) and \(AB\) respectively. What is the perimeter of triangle \(DEF\)?


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