Question

applying the side-splitter theorem
what is the length of \\(\overline{gd}\\)?
\\(gd = \square\\)
(figure with points e, g, d and f, h, d; eg = 6, fh = 8, hd = 14, and markings indicating parallel segments)

Explanation:

Step1: Apply Side - Splitter Theorem

The Side - Splitter Theorem states that if a line is parallel to one side of a triangle and intersects the other two sides, then it divides those sides proportionally. In this case, since \(GH\parallel EF\), we have \(\frac{EG}{GD}=\frac{FH}{HD}\). Wait, actually, let's correct the segments. Let's assume the triangle is \( \triangle EFD\), and \(GH\) is parallel to \(EF\). So the segments on \(ED\) are \(EG = 6\) and \(GD=x\) (what we need to find), and on \(FD\) are \(FH = 8\) and \(HD = 14\)? Wait, no, maybe the correct proportion is \(\frac{EG}{GD}=\frac{FH}{HD}\)? Wait, no, the Side - Splitter Theorem is \(\frac{EG}{GD}=\frac{FH}{HD}\) when \(GH\parallel EF\). Wait, let's re - examine the diagram. The two marked sides (with the tick marks) mean that \(GH\) is parallel to \(EF\). So the ratio of the segments on one side of the triangle is equal to the ratio of the segments on the other side. So \(\frac{EG}{GD}=\frac{FH}{HD}\). Wait, \(EG = 6\), \(FH=8\), \(HD = 14\)? No, that can't be. Wait, maybe the segments are \(EG = 6\), \(GD=x\), \(FH = 8\), and \(HD=14\)? Wait, no, perhaps the correct proportion is \(\frac{EG}{GD}=\frac{FH}{HD}\) is wrong. Let's think again. The Side - Splitter Theorem: If a line parallel to one side of a triangle intersects the other two sides, then it divides those sides into segments of proportional length. So in triangle \(EFD\), with \(GH\parallel EF\), the line \(GH\) intersects \(ED\) at \(G\) and \(FD\) at \(H\). So \(\frac{EG}{GD}=\frac{FH}{HD}\). Wait, \(EG = 6\), \(FH = 8\), \(HD=14\)? No, that would give \(\frac{6}{x}=\frac{8}{14}\), but that would be incorrect. Wait, maybe the segments are \(EG = 6\), \(GD=x\), \(FH = 8\), and \(HD = 14\) is wrong. Wait, maybe the length of \(FD\) is \(FH + HD=8 + 14=22\)? No, that doesn't seem right. Wait, perhaps I mixed up the segments. Let's assume that \(EG = 6\), \(FH = 8\), and we need to find \(GD\) such that \(\frac{EG}{GD}=\frac{FH}{HD}\), but \(HD = 14\)? No, maybe the correct proportion is \(\frac{EG}{FH}=\frac{GD}{HD}\). Let's try that. So \(\frac{6}{8}=\frac{x}{14}\). Then cross - multiply: \(8x=6\times14\), \(8x = 84\), \(x=\frac{84}{8}=\frac{21}{2}=10.5\)? No, that doesn't seem right. Wait, maybe the diagram has \(EG = 6\), \(GD=x\), \(FH = 8\), and \(HD = 14\) is incorrect. Wait, maybe the length of \(FH\) is 8 and \(HD\) is 14, and \(EG = 6\), \(GD=x\). Wait, no, perhaps the correct ratio is \(\frac{EG}{FH}=\frac{GD}{HD}\). Let's check the formula again. The Side - Splitter Theorem: \(\frac{EG}{GD}=\frac{FH}{HD}\) is equivalent to \(\frac{EG}{FH}=\frac{GD}{HD}\) by cross - multiplying. So if \(EG = 6\), \(FH = 8\), \(HD = 14\), then \(\frac{6}{8}=\frac{x}{14}\), \(x=\frac{6\times14}{8}=\frac{84}{8}=10.5\). But that seems odd. Wait, maybe the segments are \(EG = 6\), \(GD=x\), \(FH = 8\), and \(HD = 14\) is wrong. Wait, maybe the length of \(FD\) is \(FH=8\) and \(HD = 14\), so \(FD=8 + 14 = 22\), and \(ED=EG + GD=6 + x\). Then by the Side - Splitter Theorem, \(\frac{EG}{ED}=\frac{FH}{FD}\), so \(\frac{6}{6 + x}=\frac{8}{22}\). Cross - multiply: \(8(6 + x)=6\times22\), \(48+8x = 132\), \(8x=132 - 48=84\), \(x=\frac{84}{8}=10.5\). Wait, but 10.5 is \( \frac{21}{2}\). But maybe I made a mistake in identifying the segments. Wait, perhaps the correct proportion is \(\frac{EG}{GD}=\frac{FH}{HD}\), so \(EG = 6\), \(FH = 8\), \(HD = 14\), then \(\frac{6}{x}=\frac{8}{14}\), \(8x=6\times14\), \(x=\frac{84}{8}=10.5\). So \(GD = 10.5\) or \(\frac{21}{2}\). Wait, but maybe the diagram has \(FH = 8\), \(HD = 14\), \(EG = 6\), and we need to find \(GD\). So using the Side - Splitter Theorem \(\frac{EG}{GD}=\frac{FH}{HD}\), so \(GD=\frac{EG\times HD}{FH}=\frac{6\times14}{8}=\frac{84}{8}=10.5\).

Step2: Calculate the value of \(GD\)

We have the proportion from the Side - Splitter Theorem: \(\frac{EG}{GD}=\frac{FH}{HD}\). Substituting \(EG = 6\), \(FH = 8\), and \(HD = 14\) into the proportion \(\frac{6}{GD}=\frac{8}{14}\). Cross - multiply: \(8\times GD=6\times14\). Then \(8\times GD = 84\). Divide both sides by 8: \(GD=\frac{84}{8}=\frac{21}{2}=10.5\). Wait, but maybe the segments are different. Wait, maybe \(FH = 8\) and \(FD=8 + 14 = 22\), and \(EG = 6\), \(ED=6 + GD\). Then by the Side - Splitter Theorem \(\frac{EG}{ED}=\frac{FH}{FD}\), so \(\frac{6}{6 + GD}=\frac{8}{22}\). Cross - multiply: \(8\times(6 + GD)=6\times22\), \(48+8GD = 132\), \(8GD=132 - 48 = 84\), \(GD=\frac{84}{8}=10.5\). So the length of \(GD\) is \(10.5\) or \(\frac{21}{2}\).

Answer:

\(10.5\) (or \(\frac{21}{2}\))