Question

1. point m is the midpoint of \\( \overline { n l } \\). \\( j k = 8 x - 10 \\). \\( k l = 5 x + 5 \\). \\( l m = 6 x - 15 \\). \\( m n = 4 x + 5 \\). solve for x. find jk, kl, lm and mn. (9 marks)

Explanation:

Step1: Use the mid - point property

Since \(M\) is the mid - point of \(\overline{NL}\), then \(LM = MN\).
Set up the equation: \(6x−15 = 4x + 5\).
Subtract \(4x\) from both sides: \(6x-4x−15=4x - 4x+ 5\), which simplifies to \(2x−15 = 5\).
Add \(15\) to both sides: \(2x−15 + 15=5 + 15\), so \(2x=20\).
Divide both sides by \(2\): \(x=\frac{20}{2}=10\).

Step2: Calculate \(JK\)

Substitute \(x = 10\) into \(JK=8x−10\).
\(JK=8\times10−10=80 - 10=70\).

Step3: Calculate \(KL\)

Substitute \(x = 10\) into \(KL = 5x+5\).
\(KL=5\times10 + 5=50+5 = 55\).

Step4: Calculate \(LM\)

Substitute \(x = 10\) into \(LM=6x−15\).
\(LM=6\times10−15=60 - 15=45\).

Step5: Calculate \(MN\)

Substitute \(x = 10\) into \(MN=4x + 5\).
\(MN=4\times10+5=40 + 5=45\).

Answer:

\(x = 10\), \(JK = 70\), \(KL = 55\), \(LM = 45\), \(MN = 45\)