Question

applying the triangle inequality theorems and the converse of the pythagorean theorem. classify each of the triangles as acute, obtuse, or right. triangle jkl is triangle. triangle xyz is triangle. an acute an obtuse a right

Explanation:

Step1: Recall the converse of Pythagorean theorem

Let \(a\), \(b\), and \(c\) be the side - lengths of a triangle with \(c\) being the longest side. If \(a^{2}+b^{2}=c^{2}\), the triangle is a right - triangle; if \(a^{2}+b^{2}>c^{2}\), the triangle is acute; if \(a^{2}+b^{2}

Step2: Analyze \(\triangle JKL\)

In \(\triangle JKL\), \(a = 3\), \(b = 4\), \(c = 6\). Calculate \(a^{2}+b^{2}\) and \(c^{2}\). \(a^{2}+b^{2}=3^{2}+4^{2}=9 + 16=25\), and \(c^{2}=6^{2}=36\). Since \(25<36\) (i.e., \(a^{2}+b^{2}

Step3: Analyze \(\triangle XYZ\)

In \(\triangle XYZ\), \(a = 4\), \(b = 4\), \(c = 5\). Calculate \(a^{2}+b^{2}\) and \(c^{2}\). \(a^{2}+b^{2}=4^{2}+4^{2}=16 + 16=32\), and \(c^{2}=5^{2}=25\). Since \(32>25\) (i.e., \(a^{2}+b^{2}>c^{2}\)), \(\triangle XYZ\) is an acute triangle.

Answer:

Triangle JKL is an obtuse triangle.
Triangle XYZ is an acute triangle.