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 isosceles equilateral scalene

Question

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 isosceles equilateral scalene

UpStudy AI · Accepted Answer

1. Compute the side lengths:
   - For $$\overline{LM}$$ between $$L(-2,4)$$ and $$M(3,2)$$:
     $$
     LM = \sqrt{(3 - (-2))^2 + (2 - 4)^2} = \sqrt{5^2 + (-2)^2} = \sqrt{25 + 4} = \sqrt{29}.
     $$
   - For $$\overline{LN}$$ between $$L(-2,4)$$ and $$N(1,-3)$$:
     $$
     LN = \sqrt{(1 - (-2))^2 + (-3 - 4)^2} = \sqrt{3^2 + (-7)^2} = \sqrt{9 + 49} = \sqrt{58}.
     $$
   - For $$\overline{MN}$$ between $$M(3,2)$$ and $$N(1,-3)$$:
     $$
     MN = \sqrt{(1-3)^2 + (-3-2)^2} = \sqrt{(-2)^2 + (-5)^2} = \sqrt{4+25} = \sqrt{29}.
     $$

2. Check for an isosceles triangle:
   - Since $$LM = MN = \sqrt{29}$$ and $$LN = \sqrt{58}$$, the triangle has two equal sides.
   - Therefore, $$\triangle LMN$$ is isosceles.

3. Check for a right triangle:
   - Identify the longest side: $$LN = \sqrt{58}$$.
   - Verify the Pythagorean theorem:
     $$
     LM^2 + MN^2 = (\sqrt{29})^2 + (\sqrt{29})^2 = 29 + 29 = 58.
     $$
     $$
     LN^2 = (\sqrt{58})^2 = 58.
     $$
   - Since $$LM^2 + MN^2 = LN^2$$, the triangle is right.

4. Check other properties:
   - The triangle is not obtuse because it is right.
   - It is not equilateral because not all vertices have equal distance.
   - It is not scalene because two sides are congruent.

The correct descriptions are:

- $$ \text{right} $$
- $$ \text{isosceles} $$
The triangle is right and isosceles.

Select all the correct answers.
has vertices , and . Which describe the triangle?
right
obtuse
isosceles
equilateral
scalene

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Select all the correct answers. has vertices , and . Which describe the triangle? right obtuse isosceles equilateral scalene