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: Identify coordinates

From the graph, coordinates are: \( M(-4, -1) \), \( L(-4, 4) \), \( N(2, -1) \).

Step2: Check \( \overline{LM} \perp \overline{MN} \)

- \( \overline{LM} \): vertical line (same x-coordinate), slope undefined.
- \( \overline{MN} \): horizontal line (same y-coordinate), slope \( 0 \).
- Vertical and horizontal lines are perpendicular. So \( \overline{LM} \perp \overline{MN} \) is true.

Step3: Check triangle type (right triangle)

Since \( \overline{LM} \perp \overline{MN} \), \( \angle M = 90^\circ \). So it's a right triangle.

Step4: Check side lengths

- \( LM \): distance \( |4 - (-1)| = 5 \) (vertical distance).
- \( MN \): distance \( |2 - (-4)| = 6 \) (horizontal distance).
- \( LN \): distance \( \sqrt{(2 - (-4))^2 + (-1 - 4)^2} = \sqrt{36 + 25} = \sqrt{61} \).
- Two sides (LM=5, MN=6) are not equal, LN is different. So not isosceles/equilateral. Scalene? Wait, no: wait, LM=5, MN=6, LN=√61≈7.81. Wait, but wait: Wait, LM is vertical from (-4,-1) to (-4,4): length 5. MN is horizontal from (-4,-1) to (2,-1): length 6. Then LN: distance between (-4,4) and (2,-1): \( \sqrt{(2+4)^2 + (-1-4)^2} = \sqrt{36 +25}=\sqrt{61} \). So sides are 5,6,√61. So all sides different? Wait, but wait, is it isosceles? No. But wait, the right angle is at M, so it's a right triangle. Also, \( \overline{LM} \perp \overline{MN} \) is true. Let's recheck:

- \( \overline{LM} \) is vertical (x=-4), \( \overline{MN} \) is horizontal (y=-1), so they are perpendicular. So \( \overline{LM} \perp \overline{MN} \) is true.
- Right triangle: since \( \angle M = 90^\circ \), true.
- Scalene: all sides different (5,6,√61), so scalene? Wait, but the options: "the triangle is scalene" – but wait, let's check again. Wait, LM length: from y=-1 to y=4: 5 units. MN: from x=-4 to x=2: 6 units. LN: as above, √61≈7.81. So all sides different, so scalene? But also, isosceles? No. Equilateral? No. So:

- \( \overline{LM} \perp \overline{MN} \): true.
- Scalene: true? Wait, no, wait: wait, maybe I made a mistake. Wait, LM is vertical, length 5 (from y=-1 to y=4: 5). MN is horizontal, length 6 (from x=-4 to x=2: 6). Then LN: distance between (-4,4) and (2,-1): \( \sqrt{(6)^2 + (-5)^2} = \sqrt{36 +25} = \sqrt{61} \approx 7.81 \). So sides are 5,6,√61. All different, so scalene. But also, right triangle (since right angle at M). Wait, but the options:

Options:

- \( \overline{LM} \perp \overline{MN} \): true.
- Scalene: true? Wait, no, wait: wait, maybe the triangle is isosceles? Wait, no, 5,6,√61: all different. Wait, but maybe I miscalculated LM. Wait, M is at (-4, -1), L is at (-4,4). So vertical distance: 4 - (-1) = 5. Correct. MN: M(-4,-1) to N(2,-1): horizontal distance: 2 - (-4) = 6. Correct. LN: distance between (-4,4) and (2,-1): \( \sqrt{(2 - (-4))^2 + (-1 - 4)^2} = \sqrt{6^2 + (-5)^2} = \sqrt{36 +25} = \sqrt{61} \). Correct. So sides 5,6,√61: all different. So scalene? But also, right triangle (because LM and MN are perpendicular). So:

True statements:

1. \( \overline{LM} \perp \overline{MN} \) (because vertical and horizontal lines are perpendicular).
2. The triangle is scalene (all sides have different lengths: 5, 6, \( \sqrt{61} \)).
3. The triangle is a right triangle (since \( \angle M = 90^\circ \) as \( \overline{LM} \perp \overline{MN} \)).

Wait, but wait: is it isosceles? No, because no two sides are equal. Equilateral? No. So the correct options are:

- \( \overline{LM} \perp \overline{MN} \)
- The triangle is scalene
- The triangle is a right triangle

Wait, but let's confirm again. Let's check the slopes:

- Slope of LM: undefined (vertical line).
- Slope of MN: 0 (horizontal line).
- Product of slopes: undefined * 0? Wait, no, vertical and horizontal lines are perpendicular (since one is vertical, one is horizontal, angle between them is 90 degrees). So \( \overline{LM} \perp \overline{MN} \) is true.

Right triangle: yes, because it has a right angle at M.

Scalene: yes, because all sides are of different lengths (5, 6, \( \sqrt{61} \)).

Isosceles: no, because no two sides are equal.

Equilateral: no, because all sides are not equal.

So the correct options are:

- \( \overline{LM} \perp \overline{MN} \)
- The triangle is scalene
- The triangle is a right triangle

Wait, but let me check the side lengths again. LM: 5, MN: 6, LN: √61 ≈7.81. So all different, so scalene. Correct.

Answer:

• \( \overline{LM} \perp \overline{MN} \)
• The triangle is scalene
• The triangle is a right triangle