Select all the correct answers. $$ riangle \mathrm{LMN} $$ has vertices $$ \mathrm{L}(-2,4), \mathrm{M}(3,2) $$, and $$ \mathrm{N}(1,-3) $$. Which describe the triangle? right obtuse acute isosceles equilateral scalene

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Select all the correct answers. $$ 	riangle \mathrm{LMN} $$ has vertices $$ \mathrm{L}(-2,4), \mathrm{M}(3,2) $$, and $$ \mathrm{N}(1,-3) $$. Which describe the triangle? right obtuse acute isosceles equilateral scalene

UpStudy AI · Accepted Answer

1. Compute the side lengths for $$\triangle LMN$$.
   - For $$LM$$:
     $$
     LM = \sqrt{(3 - (-2))^2 + (2 - 4)^2} = \sqrt{5^2 + (-2)^2} = \sqrt{25+4} = \sqrt{29}.
     $$
   - For $$LN$$:
     $$
     LN = \sqrt{(1 - (-2))^2 + (-3 - 4)^2} = \sqrt{3^2 + (-7)^2} = \sqrt{9+49} = \sqrt{58}.
     $$
   - For $$MN$$:
     $$
     MN = \sqrt{(1-3)^2 + (-3-2)^2} = \sqrt{(-2)^2 + (-5)^2} = \sqrt{4+25} = \sqrt{29}.
     $$

2. Check for isosceles or scalene:
   - Since $$LM = MN = \sqrt{29}$$ and $$LN = \sqrt{58}$$, two sides are equal.
   - Thus, the triangle is **isosceles** (and not scalene).

3. Determine if the triangle is right:
   - A triangle is right if the square of the longest side equals the sum of the squares of the other two sides.
   - Here, the longest side is $$LN$$ with:
     $$
     LN^2 = 58.
     $$
   - The sum of the squares of the two equal sides is:
     $$
     (\sqrt{29})^2+(\sqrt{29})^2 = 29+29=58.
     $$
   - Since $$29+29=58$$, the triangle satisfies the Pythagorean theorem and is **right**.

4. Conclusion:
   - The triangle is **right** and **isosceles**.
   - It is neither obtuse, acute, equilateral, nor scalene.

Thus, the correct answers are: right, isosceles.
The triangle is right and isosceles.

Select all the correct answers. \( \triangle \mathrm{LMN} \) has vertices \( \mathrm{L}(-2,4), \mathrm{M}(3,2) \), and \( \mathrm{N}(1,-3) \). Which describe the triangle? right obtuse acute isosceles equilateral scalene

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