Question

deriving a tangent ratio for special triangles
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: Identify triangle type

Triangle LMN is a 45-45-90 right triangle (right-angled at N, two 45° angles), so it is isosceles with legs of equal length.

Step2: Determine side lengths

Given leg $LN = x$, the other leg $NM = x$. The hypotenuse $LM = x\sqrt{2}$ (by 45-45-90 triangle rule: hypotenuse = leg $\times\sqrt{2}$).

Step3: Calculate $\tan(45^\circ)$

Tangent is $\frac{\text{opposite}}{\text{adjacent}}$. For $\angle L=45^\circ$, $\tan(45^\circ)=\frac{NM}{LN}=\frac{x}{x}=1$.

Answer:

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