Question

Quadrilateral LMNO has vertices at L(-9,6),M(-1,10),N(4,0), and O(-4,-4). Is LMNO a rectangle? Justify your answer.

Answer 1

To determine if quadrilateral LMNO is a rectangle, we need to check if its opposite sides are parallel and if its adjacent sides are perpendicular. By calculating the slopes of the sides and checking their values, we can conclude that LMNO is a rectangle.

Step-by-step explanation:

The given quadrilateral LMNO has vertices at L(-9,6), M(-1,10), N(4,0), and O(-4,-4). To determine if it is a rectangle, we need to check if its opposite sides are parallel and if its adjacent sides are perpendicular. Step by step:

• Step 1: Calculate the slopes of LM and NO. The slope of LM = (10-6) / (-1-(-9)) = 4 / 8 = 1/2. The slope of NO = (0-(-4)) / (4-(-4)) = 4/8 = 1/2.
• Step 2: Calculate the slopes of LO and MN. The slope of LO = (-4-6) / (-4-(-9)) = -10 / 5 = -2. The slope of MN = (0-10) / (4-(-1)) = -10 / 5 = -2.
• Step 3: Check if opposite sides are parallel. The slopes of LM and NO are the same, and the slopes of LO and MN are the same. Therefore, the opposite sides of LMNO are parallel.
• Step 4: Check if adjacent sides are perpendicular. The product of the slopes of LM and LO is (1/2) * (-2) = -1. The product of the slopes of NO and MN is (1/2) * (-2) = -1. Therefore, the adjacent sides of LMNO are perpendicular.
• Step 5: Since the opposite sides of LMNO are parallel and the adjacent sides are perpendicular, we can conclude that LMNO is indeed a rectangle.