To determine the most precise name for the quadrilateral ABCD with vertices A(3, 2), B(1, 4), C(4, 4), and D(2, 2), we follow a systematic approach involving basic geometry principles.
First, let’s plot the points to visualize the shape:
- A(3, 2)
- B(1, 4)
- C(4, 4)
- D(2, 2)
Next, we calculate the lengths of the sides and the slopes of the lines connecting these points:
- Length AB: √[(1 – 3)² + (4 – 2)²] = √[4 + 4] = √8 = 2√2
- Length BC: √[(4 – 1)² + (4 – 4)²] = √[9] = 3
- Length CD: √[(4 – 2)² + (4 – 2)²] = √[4 + 4] = √8 = 2√2
- Length DA: √[(3 – 2)² + (2 – 2)²] = √[1] = 1
Next, we need to check the slopes to see if any sides are parallel:
- Slope AB: (4 – 2)/(1 – 3) = 2/(-2) = -1
- Slope BC: (4 – 4)/(4 – 1) = 0
- Slope CD: (4 – 2)/(4 – 2) = 2/2 = 1
- Slope DA: (2 – 2)/(3 – 2) = 0
The slopes tell us:
- AB is not parallel to CD since their slopes are -1 and 1.
- AB is not parallel to DA since (-1) is not equal to 0.
- BC is parallel to DA since both slopes are 0.
This tells us that ABCD has one pair of opposite sides that are parallel.
Additionally, the opposite sides have equal lengths:
- AB = CD (both are 2√2)
- BC = DA (3 and 1 are not equal).
Thus, we can conclude that ABCD is a trapezoid as it has at least one pair of parallel sides (AB and CD).
Therefore, the most precise name for this quadrilateral ABCD is a trapezoid.