Question

what is the length of segment lm? units

Explanation:

Step1: Set up the equation

Since \(NK = NM\) (marked as equal), we have \(14x - 3=25\).
Solve for \(x\):
\(14x=25 + 3\)
\(14x=28\)
\(x = 2\)

Step2: Find the length of \(KL\)

Substitute \(x = 2\) into \(9x+5\):
\(9\times2+5=18 + 5=23\)

Step3: Use the Pythagorean theorem in \(\triangle NLK\) (right - triangle at \(L\))

Since \(NK = NM = 25\) (from step 1), and \(KL = 23\) (from step 2).
Let \(NL = h\), then \(h=\sqrt{NK^{2}-KL^{2}}=\sqrt{25^{2}-23^{2}}=\sqrt{(25 + 23)(25 - 23)}=\sqrt{48\times2}=\sqrt{96}=4\sqrt{6}\) (but we can also note that since \(NL\) is the perpendicular bisector, \(LM=KL\) when considering the properties of congruent triangles (by SSS, \(\triangle NLK\cong\triangle NLM\) as \(NK = NM\), \(NL=NL\), and \(\angle NLK=\angle NLM = 90^{\circ}\)). So \(LM=KL\)

Another way:
Since \(NK = NM = 25\) (given as equal segments), and \(NL\) is perpendicular to \(KM\) (right - angle at \(L\)), by the property of congruent right - triangles (\(\triangle NLK\) and \(\triangle NLM\) (HL - Hypotenuse - Leg congruence: \(NK = NM\), \(NL = NL\)), so \(KL=LM\)
Substitute \(x = 2\) into \(9x + 5\): \(9(2)+5=18 + 5=23\)

Answer:

\(23\)