Question

question 4 (multiple choice worth 1 points) (05.01r mc) in the figure below, the length of segment cb is 58 units and the length of segment bg is 120 units. what is the length of segment ge? 62\sqrt{3} 116 58\sqrt{2} 124

Explanation:

Step1: Find the length of \( AB \)

In right - triangle \( ABC \), \(\angle B = 90^{\circ}\), \(\angle BAC=45^{\circ}\), so \(AB = BC\) (since in a \(45 - 45-90\) triangle, the legs are equal). Given \(BC = 58\) units, then \(AB = 58\) units.

Step2: Find the length of \( AG \)

Since \(BG=120\) units and \(AB = 58\) units, then \(AG=BG - AB\). So \(AG=120 - 58=62\) units.

Step3: Find the length of \( GE \)

In right - triangle \( AGE \), \(\angle G = 90^{\circ}\), \(\angle EAG = 60^{\circ}\). We know that \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\) is not relevant here. Using the relationship in a \(30 - 60-90\) triangle: if the side adjacent to the \(60^{\circ}\) angle (\(AG\)) is \(x\), and the side opposite to the \(60^{\circ}\) angle (\(GE\)) is \(x\sqrt{3}\) (where in a \(30 - 60-90\) triangle, sides are in the ratio \(1:\sqrt{3}:2\), and if the shorter leg (adjacent to \(60^{\circ}\)) is \(a\), the longer leg (opposite to \(60^{\circ}\)) is \(a\sqrt{3}\)). Here \(a = AG = 62\) units. So \(GE=62\sqrt{3}\) units.

Answer:

\(62\sqrt{3}\)