Question

what is true about triangle lmn? check all that apply.

□ lm ⊥ mn.

□ the triangle is scalene.

□ the triangle is equilateral.

□ the triangle is isosceles.

□ the triangle is a right triangle.

Explanation:

Step1: Calculate the lengths of the sides

Use the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
For \(LM\): \(L(-3,4)\), \(M(-3,-1)\), \(d_{LM}=\sqrt{(-3 + 3)^2+(4 + 1)^2}=5\).
For \(MN\): \(M(-3,-1)\), \(N(2,-1)\), \(d_{MN}=\sqrt{(2 + 3)^2+(-1+1)^2}=5\).
For \(LN\): \(L(-3,4)\), \(N(2,-1)\), \(d_{LN}=\sqrt{(2 + 3)^2+(-1 - 4)^2}=\sqrt{25 + 25}=5\sqrt{2}\).

Step2: Check the perpendicularity

The slope of \(LM\) is undefined (vertical line \(x=-3\)). The slope of \(MN\) is \(0\) (horizontal line \(y = - 1\)). A vertical and a horizontal line are perpendicular, so \(LM\perp MN\).

Step3: Classify the triangle

Since \(LM = MN = 5\), the triangle is isosceles. Also, because \(LM\perp MN\), it is a right - triangle.

Answer:

\(\overline{LM}\perp\overline{MN}\), The triangle is isosceles, The triangle is a right triangle.