Question

The coordinates of the vertices of △ LMN are L (-2, 4), M (3, -1), and N (0, -4). Determine whether △ LMN is a right triangle and support your decision.

Answer 1

By calculating the side lengths of △ LMN using the distance formula and finding that LM² + MN² equals NL², it has been determined that △ LMN is a right triangle.

Step-by-step explanation:

To determine whether △ LMN is a right triangle, we can use the distance formula to find the lengths of each side and then apply the Pythagorean theorem.

The distance formula is √((x2-x1)² + (y2-y1)²). Applying this to the given coordinates:

• LM = √((3-(-2))² + (-1-4)²) = √(25 + 25) = √50
• MN = √((0-3)² + (-4-(-1))²) = √(9 + 9) = √18
• NL = √((-2-0)² + (4-(-4))²) = √(4 + 64) = √68

Now, we square the lengths to see if they follow Pythagoras' rule for a right triangle (a² + b² = c²), where c is the hypotenuse.

• (LM)² = 50
• (MN)² = 18
• (NL)² = 68

50 + 18 = 68, so LM² + MN² = NL². Thus, △ LMN is a right triangle.