Question

applying the side-splitter theorem
what is the length of (overline{gd})?
(gd = square)
(diagram: triangle with points e, f, d. segment (eg = 6), (fh = 8), (hd = 14). arrows on (ef) and (gh) indicate parallel lines.)

Explanation:

Step1: Recall 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 triangle \(EFD\), since \(GH\parallel EF\), we have \(\frac{EG}{GD}=\frac{FH}{HD}\). Wait, actually, the correct proportion from the Side - Splitter Theorem is \(\frac{EG}{GD}=\frac{FH}{HD}\)? No, let's correct that. The Side - Splitter Theorem: If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally. So in \(\triangle EFD\), with \(GH\parallel EF\), the segments on \(ED\) are \(EG = 6\) and \(GD=x\) (let \(GD=x\)), and on \(FD\) are \(FH = 8\) and \(HD = 14\). Wait, no, the correct proportion is \(\frac{EG}{GD}=\frac{FH}{HD}\)? Wait, no, the sides are \(ED\) and \(FD\). Wait, actually, the theorem is \(\frac{EG}{GD}=\frac{FH}{HD}\)? Wait, no, let's label the triangle properly. Let \(E\), \(G\), \(D\) be on one side, and \(F\), \(H\), \(D\) be on the other side, with \(GH\parallel EF\). So the proportion should be \(\frac{EG}{GD}=\frac{FH}{HD}\)? Wait, no, the correct formula is \(\frac{EG}{GD}=\frac{FH}{HD}\)? Wait, no, let's think again. The Side - Splitter Theorem: If a line parallel to \(EF\) (in \(\triangle EFD\)) intersects \(ED\) at \(G\) and \(FD\) at \(H\), then \(\frac{EG}{GD}=\frac{FH}{HD}\). Wait, given \(EG = 6\), \(FH=8\), \(HD = 14\), and we need to find \(GD\) (let \(GD=x\)). So according to the theorem, \(\frac{EG}{GD}=\frac{FH}{HD}\) → \(\frac{6}{x}=\frac{8}{14}\). Wait, no, that seems reversed. Wait, maybe the correct proportion is \(\frac{EG}{ED}=\frac{FH}{FD}\), but \(ED=EG + GD=6 + x\) and \(FD=FH + HD=8 + 14 = 22\). Wait, no, I think I mixed up the segments. Let's re - express: The Side - Splitter Theorem says that \(\frac{EG}{GD}=\frac{FH}{HD}\) when \(GH\parallel EF\). Wait, no, actually, the correct proportion is \(\frac{EG}{GD}=\frac{FH}{HD}\)? Wait, let's check the lengths. If \(FH = 8\) and \(HD = 14\), and \(EG = 6\), let \(GD=x\). Then by the Side - Splitter Theorem, \(\frac{EG}{GD}=\frac{FH}{HD}\) → \(\frac{6}{x}=\frac{8}{14}\). Cross - multiplying: \(8x=6\times14\) → \(8x = 84\) → \(x=\frac{84}{8}=\frac{21}{2}=10.5\)? Wait, that can't be right. Wait, maybe the proportion is \(\frac{EG}{ED}=\frac{FH}{FD}\), where \(ED = EG+GD=6 + x\) and \(FD=FH + HD=8 + 14 = 22\). Then \(\frac{6}{6 + x}=\frac{8}{22}\). Cross - multiplying: \(8(6 + x)=6\times22\) → \(48+8x = 132\) → \(8x=132 - 48=84\) → \(x=\frac{84}{8}=\frac{21}{2}=10.5\). Wait, but maybe I had the proportion reversed. Wait, the Side - Splitter Theorem: If a line is parallel to one side of a triangle and intersects the other two sides, then it divides those sides proportionally. So the ratio of the segments on one side is equal to the ratio of the segments on the other side. So if \(GH\parallel EF\), then \(\frac{EG}{GD}=\frac{FH}{HD}\). Wait, \(EG = 6\), \(FH = 8\), \(HD = 14\), so \(\frac{6}{GD}=\frac{8}{14}\). Solving for \(GD\): \(GD=\frac{6\times14}{8}=\frac{84}{8}=\frac{21}{2}=10.5\). Wait, but let's check again. Maybe the triangle is labeled such that \(EF\) is parallel to \(GH\), and \(F\), \(H\), \(D\) are on one side with \(FH = 8\), \(HD = 14\), and \(E\), \(G\), \(D\) are on the other side with \(EG = 6\), \(GD=x\). Then by the Side - Splitter Theorem, \(\frac{EG}{GD}=\frac{FH}{HD}\) → \(\frac{6}{x}=\frac{8}{14}\) → \(x=\frac{6\times14}{8}=\frac{84}{8}=10.5\). So \(GD = 10.5\) or \(\frac{21}{2}\).

Step1: Identify the 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. For \(\triangle EFD\) with \(GH\parallel EF\), we have \(\frac{EG}{GD}=\frac{FH}{HD}\).

Step2: Substitute the known values

We know \(EG = 6\), \(FH = 8\), and \(HD = 14\). Let \(GD=x\). Substituting into the proportion: \(\frac{6}{x}=\frac{8}{14}\).

Step3: Solve for \(x\) (GD)

Cross - multiply to get \(8x=6\times14\).
Calculate \(6\times14 = 84\), so \(8x = 84\).
Divide both sides by 8: \(x=\frac{84}{8}=\frac{21}{2}=10.5\).

Answer:

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