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 side: \( 2x + 4x - 10 = 4x \) → \( 6x - 10 = 4x \). Subtract \( 4x \) from both sides: \( 2x - 10 = 0 \). Add 10 to both sides: \( 2x = 10 \). Divide by 2: \( x = 5 \).

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

Substitute \( x = 5 \) into \( IJ = 4x - 10 \): \( IJ = 4(5) - 10 = 20 - 10 = 10 \).

Answer:

10