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 right isosceles triangle (two 45° angles, right angle at N). In such triangles, the legs are equal. Given $LN = x$, so $NM = x$.

Step2: Calculate hypotenuse length

Use Pythagorean theorem for hypotenuse $LM$:
$$LM = \sqrt{LN^2 + NM^2} = \sqrt{x^2 + x^2} = \sqrt{2x^2} = x\sqrt{2}$$

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

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

Answer:

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