Question

1. which of the following equations is false?

triangle diagram with vertices j, k, l; m is midpoint of jk, lm perpendicular to jk, angles at l: 30° and 30°, lk = 48
options:
○ lm = 24
○ jm = 24
○ jl = 48
○ jk = 48

Explanation:

Step1: Analyze triangle properties

In triangle \(JLK\), \(LM\perp JK\), \(\angle JLM = \angle KLM = 30^\circ\), and \(LK = 48\). Triangles \(JLM\) and \(KLM\) are congruent (by ASA: right angle, \(LM\) common, \(\angle JLM=\angle KLM\)). So \(JL = LK = 48\), \(JM = KM\). In right triangle \(KLM\), \(\cos(30^\circ)=\frac{LM}{LK}\), \(\sin(30^\circ)=\frac{KM}{LK}\). \(\sin(30^\circ)=\frac{1}{2}\), so \(KM=\frac{1}{2}\times48 = 24\), so \(JM = 24\), \(JL = 48\), \(JK=JM + KM=24 + 24 = 48\)? Wait, no, wait: Wait, in right triangle \(KLM\), \(\angle KLM = 30^\circ\), hypotenuse \(LK = 48\), so the side opposite \(30^\circ\) is \(KM\), so \(KM=\frac{1}{2}\times LK = 24\), and \(LM = LK\times\cos(30^\circ)=48\times\frac{\sqrt{3}}{2}=24\sqrt{3}\approx41.57\), not 24. So \(LM = 24\) is false. Let's check each option:

• \(LM = 24\): As above, \(LM = 48\times\cos(30^\circ)=24\sqrt{3}

eq24\), so this is false.

• \(JM = 24\): \(JM = KM = 24\) (since \(KM = \frac{1}{2}LK = 24\) as \(KM\) is opposite \(30^\circ\) in right triangle \(KLM\)), so true.
• \(JL = 48\): \(JL = LK = 48\) (triangle congruence), true.
• \(JK = 48\): \(JK = JM + KM = 24 + 24 = 48\), true.

Step2: Confirm the false equation

From the analysis, \(LM = 24\) is false.

Answer:

\(LM = 24\) (the option with \(LM = 24\))