Question

in... is the midpoint? if... diagram: line segment from h to y, with parts labeled (8x - 2), (9), (7x - 6); total length labeled (3x + 9)? blank boxes for answers.

Explanation:

Step1: Analyze the segment addition

From the diagram, we know that \( HX + XY = HY \). Given \( HX = 8x - 2 \), \( XY = 7x - 6 \), and \( HY = 5x + 9 \). So we set up the equation: \( (8x - 2)+(7x - 6)=5x + 9 \).

Step2: Simplify the left - hand side

Combine like terms: \( 8x+7x-2 - 6=5x + 9\), which simplifies to \( 15x-8 = 5x + 9 \).

Step3: Solve for x

Subtract \( 5x \) from both sides: \( 15x-5x-8=5x - 5x+9\), so \( 10x-8 = 9 \). Then add 8 to both sides: \( 10x-8 + 8=9 + 8\), which gives \( 10x=17 \). Divide both sides by 10: \( x=\frac{17}{10}=1.7 \).

Step4: Find the length of HX

Substitute \( x = 1.7 \) into \( HX=8x - 2 \). \( HX=8\times1.7-2=13.6 - 2 = 11.6 \).

Step5: Find the length of XY

Substitute \( x = 1.7 \) into \( XY = 7x-6 \). \( XY=7\times1.7-6 = 11.9 - 6=5.9 \).

Step6: Find the length of HY

Substitute \( x = 1.7 \) into \( HY = 5x + 9 \). \( HY=5\times1.7+9=8.5 + 9 = 17.5 \). We can also check by adding \( HX+XY=11.6 + 5.9 = 17.5 \), which matches.

Answer:

If we assume the question is to find \( x \), then \( x = 1.7 \) (or \( \frac{17}{10} \)). If it's to find the lengths: \( HX = 11.6 \), \( XY = 5.9 \), \( HY = 17.5 \) (depending on the exact question, but based on the segment addition, the value of \( x \) is \( \frac{17}{10} \) or \( 1.7 \)).