Question

finding a segment length
what is the value of x and the length of segment de?

1. \\( \frac { 5 } { 9 } = \frac { 2 x + 3 } { 9 } \\)
2. \\( 10 x + 15 = 9 ( 9 ) \\)

\\( x = \\)
length of \\( \overline { d e } = \\) units

Explanation:

Step1: Set up the proportion

Since the two right - angled triangles \( \triangle CDF\) and \( \triangle EDF\) are similar (by AA similarity, as both have a right angle and share an angle), we can set up the proportion \(\frac{CD}{DF}=\frac{DF}{DE}\). Given \(CD = 5\), \(DF = 9\), and \(DE=2x + 3\), the proportion is \(\frac{5}{9}=\frac{9}{2x+3}\). Cross - multiply to get \(5(2x + 3)=9\times9\).

Step2: Expand and solve for \(x\)

Expand \(5(2x + 3)\) to \(10x+15\). So the equation becomes \(10x + 15=81\). Subtract 15 from both sides: \(10x=81 - 15\), which simplifies to \(10x=66\). Divide both sides by 10: \(x=\frac{66}{10}=6.6\).

Step3: Find the length of \(DE\)

Substitute \(x = 6.6\) into \(DE = 2x+3\). Then \(DE=2\times6.6+3\). Calculate \(2\times6.6 = 13.2\), and \(13.2+3=16.2\).

Answer:

\(x = 6.6\), Length of \(\overline{DE}=16.2\) units