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: Identify Triangle Type

Triangle \( LMN \) is a right - isosceles triangle (two angles are \( 45^{\circ} \), so the two legs are equal, \( LN = NM=x \), and the hypotenuse \( LM \) can be found using Pythagorean theorem or special right - triangle ratios.

Step2: Analyze Side Lengths

• For a \( 45 - 45 - 90 \) triangle, the ratio of the legs to the hypotenuse is \( 1:1:\sqrt{2} \). The legs \( LN = x \) and \( NM=x \) (since the two non - right angles are equal, the opposite sides are equal). The hypotenuse \( LM=\sqrt{LN^{2}+NM^{2}}=\sqrt{x^{2}+x^{2}}=\sqrt{2x^{2}} = x\sqrt{2} \).
• Analyze the tangent of \( 45^{\circ} \): In a right triangle, \( \tan\theta=\frac{\text{opposite}}{\text{adjacent}} \). For \( \theta = 45^{\circ} \), opposite and adjacent sides are equal (since it's an isosceles right triangle). So \( \tan(45^{\circ})=\frac{\text{opposite}}{\text{adjacent}} = 1 \) (because opposite = adjacent in a \( 45 - 45 - 90 \) triangle).

Step3: Evaluate Each Option

• Option 1: \( NM = x \): Since \( LN = x \) and \( \triangle LMN \) is isosceles right - triangle with \( \angle L=\angle M = 45^{\circ} \), \( NM = LN=x \), so this is true.
• Option 2: \( NM=x\sqrt{2} \): From above, \( NM = x \), so this is false.
• Option 3: \( LM=x\sqrt{2} \): Using Pythagorean theorem, \( LM=\sqrt{x^{2}+x^{2}}=x\sqrt{2} \), so this is true.
• Option 4: \( \tan(45^{\circ})=\frac{\sqrt{2}}{2} \): \( \frac{\sqrt{2}}{2}\approx0.707 \) and \( \tan(45^{\circ}) = 1 \), so this is false.
• Option 5: \( \tan(45^{\circ}) = 1 \): Since in a \( 45 - 45 - 90 \) triangle, opposite and adjacent sides are equal, \( \tan(45^{\circ})=\frac{\text{opposite}}{\text{adjacent}} = 1 \), so this is true.

Answer:

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