Question

in the diagram below, \\(\overline{gh}\\) is parallel to \\(\overline{de}\\). solve for \\(x\\). round your answer to the nearest tenth if necessary.\
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(diagram: triangle with points f, h, e at the top; f to h is 13.3, h to e is 10.7. point g is on fd, with fg = x, gd = 17.8. gh is parallel to de.)\
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answer attempt 1 out of 2\
\\(x = \\) blank submit answer

Explanation:

Step1: Identify Similar Triangles

Since \(\overline{GH} \parallel \overline{DE}\), by the Basic Proportionality Theorem (Thales' theorem), \(\triangle FGH \sim \triangle FDE\) (similar triangles). This means the ratios of corresponding sides are equal.

Step2: Set Up Proportion

The ratio of \(FH\) to \(FE\) should equal the ratio of \(FG\) to \(FD\). First, find \(FE = FH + HE = 13.3 + 10.7 = 24\). Let \(FG = x\) and \(FD = x + 17.8\). So the proportion is \(\frac{FH}{FE}=\frac{FG}{FD}\), which is \(\frac{13.3}{24}=\frac{x}{x + 17.8}\).

Step3: Solve the Proportion

Cross - multiply: \(13.3(x + 17.8)=24x\).
Expand the left side: \(13.3x+13.3\times17.8 = 24x\).
Calculate \(13.3\times17.8 = 236.74\). So the equation becomes \(13.3x + 236.74 = 24x\).
Subtract \(13.3x\) from both sides: \(236.74=24x - 13.3x\).
Simplify the right side: \(236.74 = 10.7x\).
Solve for \(x\): \(x=\frac{236.74}{10.7}\approx11.7\).

Answer:

11.7