Question

find the measure of $overline{ij}$. answer attempt 1 out of 2 submit answer

Explanation:

Step1: Apply the Law of Cosines

The Law of Cosines for a triangle with sides \(a\), \(b\), \(c\) and the angle \(\theta\) opposite to side \(c\) is \(c^{2}=a^{2}+b^{2}-2ab\cos\theta\). In \(\triangle IHJ\), let \(IH = a = 34\), \(HJ=b = 32\), and we want to find \(IJ = c\). Assume the included - angle between \(IH\) and \(HJ\) is \(\angle H\). But since no angle measure is given, if we assume \(\angle H = 90^{\circ}\) (right - triangle case, as no other information about angles is provided), then \(\cos H=\cos90^{\circ}=0\).

Step2: Calculate \(IJ\)

Using the Pythagorean theorem (a special case of the Law of Cosines when \(\cos\theta = 0\)), \(IJ^{2}=IH^{2}+HJ^{2}\). Substitute \(a = 34\) and \(b = 32\) into the formula: \(IJ^{2}=34^{2}+32^{2}=1156 + 1024=2180\). Then \(IJ=\sqrt{2180}=2\sqrt{545}\approx46.7\).

Answer:

\(2\sqrt{545}\approx46.7\)