Question

solve for x. round your answer to the nearest tenth if necessary.
(image of a triangle with points f, g, d, h, e; fg = 3.3, gd = 1.7, he = 2, fh = x, and gh parallel to de, creating similar triangles)

Explanation:

Step1: Identify Similar Triangles

Since \( GH \parallel DE \), triangles \( FGH \) and \( FDE \) are similar by the Basic Proportionality Theorem (Thales' theorem). So, the ratios of corresponding sides are equal.
The sides of \( \triangle FGH \) are \( FG = 3.3 \) and \( FH = x \).
The sides of \( \triangle FDE \) are \( FD = FG + GD = 3.3 + 1.7 = 5 \) and \( FE = FH + HE = x + 2 \).

Step2: Set Up Proportion

For similar triangles, \( \frac{FG}{FD}=\frac{FH}{FE} \)
Substitute the known values: \( \frac{3.3}{5}=\frac{x}{x + 2} \)

Step3: Cross - Multiply

Cross - multiplying gives \( 3.3(x + 2)=5x \)
Expand the left side: \( 3.3x+6.6 = 5x \)

Step4: Solve for \( x \)

Subtract \( 3.3x \) from both sides: \( 6.6=5x - 3.3x \)
Simplify the right side: \( 6.6 = 1.7x \)
Then, \( x=\frac{6.6}{1.7}\approx3.9 \)

Answer:

\( x\approx3.9 \)