Question

which statements are true regarding triangle lmn? check all that apply. □ nm = x □ nm = x√2 □ lm = x√2 □ tan(45°)=√2/2 □ tan(45°)=1

Explanation:

Step1: Recall tangent - value of 45 degrees

The formula for tangent is $\tan\theta=\frac{\sin\theta}{\cos\theta}$. For $\theta = 45^{\circ}$, $\sin45^{\circ}=\frac{\sqrt{2}}{2}$ and $\cos45^{\circ}=\frac{\sqrt{2}}{2}$. So, $\tan45^{\circ}=\frac{\sin45^{\circ}}{\cos45^{\circ}}=\frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1$.

Step2: Use right - triangle trigonometry and Pythagorean theorem

In right - triangle $LMN$ with $\angle M = 45^{\circ}$ and $\angle N=90^{\circ}$, $\angle L = 45^{\circ}$. So, it is a 45 - 45-90 right - triangle. Let $NM = a$ and $LN=a$ (since in a 45 - 45-90 triangle, the legs are equal). By the Pythagorean theorem $LM^{2}=LN^{2}+NM^{2}$. If $LN = NM=x$, then $LM=\sqrt{x^{2}+x^{2}}=\sqrt{2x^{2}}=x\sqrt{2}$.

Answer:

A. $\tan(45^{\circ}) = 1$
C. $LM=x\sqrt{2}$
D. $NM = x$