Question

point i is on line segment \\(\overline{hj}\\). given \\(hi = 2x\\), \\(hj = 4x\\), and \\(ij = 4x - 10\\), determine the numerical length of \\(\overline{ij}\\).

Explanation:

Step1: Use segment addition postulate

Since \( I \) is on \( \overline{HJ} \), we have \( HI + IJ = HJ \). Substituting the given expressions: \( 2x + (4x - 10) = 4x \).

Step2: Solve for \( x \)

Simplify the left - hand side: \( 2x+4x - 10=4x\), which becomes \( 6x - 10 = 4x \). Subtract \( 4x \) from both sides: \( 6x-4x - 10=4x - 4x\), so \( 2x-10 = 0 \). Add 10 to both sides: \( 2x=10 \). Divide both sides by 2: \( x = 5 \).

Step3: Find the length of \( \overline{IJ} \)

Substitute \( x = 5 \) into the expression for \( IJ \): \( IJ=4x - 10 \). So \( IJ = 4\times5-10=20 - 10=10 \).

Answer:

The numerical length of \( \overline{IJ} \) is \( 10 \).