Question

The vertices of quadrilateral LMNP are L(-1,-1), M(6,-2), N(5,-9), and P(-2,-8). Using the distance formula, determine the most precise classification of LMNP: parallelogram, rectangle, rhombus, or square​

Answer 1

Quadrilateral LMNP is classified as a square because it has equal-length sides and equal-length diagonals, determined using the distance formula.

Step-by-step explanation:

To determine the most precise classification of quadrilateral LMNP with vertices L(-1,-1), M(6,-2), N(5,-9), and P(-2,-8), we use the distance formula to calculate the lengths of the sides:

d = √((x2 - x1)^2 + (y2 - y1)^2)

Calculating the distances between each pair of adjacent vertices:

• LM = √((6 - (-1))^2 + (-2 - (-1))^2) = √(49 + 1) = √50
• MN = √((5 - 6)^2 + (-9 - (-2))^2) = √(1 + 49) = √50
• NP = √((5 - (-2))^2 + (-9 - (-8))^2) = √(49 + 1) = √50
• PL = √((-2 - (-1))^2 + (-8 - (-1))^2) = √(1 + 49) = √50

All sides of the quadrilateral are of equal length, which means that LMNP is either a rhombus or a square. To further classify it, we need to calculate the diagonals:

• LN = √((5 - (-1))^2 + (-9 - (-1))^2) = √(6^2 + (-8)^2) = √(36 + 64) = √100 = 10
• MP = √((6 - (-2))^2 + (-2 - (-8))^2) = √(8^2 + 6^2) = √(64 + 36) = √100 = 10

Both diagonals are also of equal length. Therefore, quadrilateral LMNP with all sides equal and diagonals equal is a square.