Show that the points A(-2, 6), B(5, 3), C(-1, -11) and D(-8, -8) are the vertices of a rectangle.
easy with vectors, but they are not usually a tool used in analytic geometry.
slope of AB is -3/7
slope of CD is -3/7, so AB||CD
slope of BC is 7/3
slope of AD is 7/3, so BC||AD
since slopes are negative reciprocals, the pairs of sides are perpendicular
So, ABCD is a rectangle.
To show that the points A(-2, 6), B(5, 3), C(-1, -11), and D(-8, -8) are the vertices of a rectangle, we need to prove two things:
1. The lengths of the sides AB, BC, CD, and DA are equal in pairs.
2. The diagonals AC and BD are equal in length and bisect each other.
Let's start by calculating the lengths of the sides:
1. Lengths of the sides:
- Side AB: √((x₂ - x₁)² + (y₂ - y₁)²)
- AB = √((5 - (-2))² + (3 - 6)²) = √(49 + 9) = √58
- Side BC: √((x₂ - x₁)² + (y₂ - y₁)²)
- BC = √((-1 - 5)² + (-11 - 3)²) = √(36 + 196) = √232
- Side CD: √((x₂ - x₁)² + (y₂ - y₁)²)
- CD = √((-8 - (-1))² + (-8 - (-11))²) = √(49 + 9) = √58
- Side DA: √((x₂ - x₁)² + (y₂ - y₁)²)
- DA = √((-2 - (-8))² + (6 - (-8))²) = √(36 + 196) = √232
Since AB = CD and BC = DA, we have shown that the lengths of the sides are equal in pairs.
2. Let's calculate the lengths of the diagonals and their midpoints:
- Length of AC: √((x₂ - x₁)² + (y₂ - y₁)²)
- AC = √((-1 - (-2))² + (-11 - 6)²) = √(1 + 289) = √290
- Length of BD: √((x₂ - x₁)² + (y₂ - y₁)²)
- BD = √((5 - (-8))² + (3 - (-8))²) = √(169 + 121) = √290
The lengths of the diagonals AC and BD are equal.
Now, let's find the midpoints of the diagonals:
- Midpoint of AC:
- x-coordinate: (x₁ + x₂)/2
- (A + C) / 2 = (-2 + (-1))/2 = -3/2
- y-coordinate: (y₁ + y₂)/2
- (6 + (-11)) / 2 = -5/2
- Midpoint of BD:
- x-coordinate: (x₁ + x₂)/2
- (B + D) / 2 = (5 + (-8))/2 = -3/2
- y-coordinate: (y₁ + y₂)/2
- (3 + (-8)) / 2 = -5/2
The midpoints of the diagonals are (-3/2, -5/2) for both AC and BD.
Since the lengths of the diagonals AC and BD are equal, and their midpoints are the same, we have shown that the diagonals bisect each other.
Therefore, based on these results, we can conclude that the points A(-2, 6), B(5, 3), C(-1, -11), and D(-8, -8) are the vertices of a rectangle.