Determine whether a figure with the given vertices is a parallelogram. Use the method indicated.
A(3, − 9), B(10, 1), C(4, 10), D(−9, 3); Distance and Slope Formulas
so parallelograms must have opposite sides congruent, in this case AB and CD, and BC and AD.
Using the distance formula:
AB=sqrt((10-3)^2+(1-(-9))^2)
sqrt(7^2+10^2)
sqrt(149)
CD=sqrt((-9-4)^2+(3-10)^2)
sqrt(-13^2+-7^2)
sqrt(169+49)
sqrt(218)
Since AB is not equal to CD, this figure cannot be a parallelogram
To determine whether a figure with the given vertices is a parallelogram, we can use the distance and slope formulas.
Step 1: Calculate the distances between the vertices AB, BC, CD, and AD.
We can use the distance formula to calculate the distance between two points (x1, y1) and (x2, y2):
Distance = √((x2 - x1)^2 + (y2 - y1)^2)
Calculating the distances:
- Distance AB = √((10-3)^2 + (1-(-9))^2)
= √(7^2 + 10^2)
= √(49 + 100)
= √149
- Distance BC = √((4-10)^2 + (10-1)^2)
= √((-6)^2 + 9^2)
= √(36 + 81)
= √117
- Distance CD = √((-9-4)^2 + (3-10)^2)
= √((-13)^2 + (-7)^2)
= √(169 + 49)
= √218
- Distance AD = √((3-(-9))^2 + (-9-3)^2)
= √((12)^2 + (-12)^2)
= √(144 + 144)
= √288
= 12√2
Step 2: Calculate the slopes of the sides AB, BC, CD, and AD.
We can use the slope formula to calculate the slope between two points (x1, y1) and (x2, y2):
Slope = (y2 - y1) / (x2 - x1)
Calculating the slopes:
- Slope AB = (1 - (-9)) / (10 - 3)
= 10 / 7
- Slope BC = (10 - 1) / (4 - 10)
= 9 / (-6)
= -3/2
- Slope CD = (3 - 10) / (-9 - 4)
= (-7) / (-13)
= 7/13
- Slope AD = (-9 - 3) / (3 - (-9))
= (-12) / (12)
= -1
Step 3: Check the conditions for parallelogram:
For a quadrilateral to be a parallelogram, the following conditions must be met:
1. Opposite sides are parallel: This means that the slopes of opposite sides are equal.
2. Opposite sides are congruent: This means that the distances between opposite sides are equal.
Checking for condition 1:
- Slope AB ≠ Slope CD (10/7 ≠ 7/13)
- Slope BC ≠ Slope AD (-3/2 ≠ -1)
Since the slopes of opposite sides are not equal, the figure is not a parallelogram.
Therefore, using the distance and slope formulas, we determined that the figure with the given vertices A(3, − 9), B(10, 1), C(4, 10), D(−9, 3) is not a parallelogram.
To determine if a figure with given vertices is a parallelogram, we can use a combination of the distance formula and slope formula.
1. Distance Formula:
The distance formula provides a way to calculate the distance between two points (x1, y1) and (x2, y2) in a coordinate plane. The formula is given by:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Using the distance formula, we can calculate the lengths of the sides of the figure formed by the given vertices.
2. Slope Formula:
The slope formula allows us to calculate the slope of a line passing through two points (x1, y1) and (x2, y2). The formula is given by:
m = (y2 - y1) / (x2 - x1)
Using the slope formula, we can determine if the opposite sides of the figure formed by the given vertices have the same slope.
To determine if the figure with the given vertices (A, B, C, D) is a parallelogram, follow these steps:
Step 1: Calculate the distances between the vertices.
- Calculate the length of side AB, BC, CD, and DA using the distance formula.
Step 2: Check if opposite sides have the same lengths.
- Compare AB and CD, and BC and DA. If AB = CD and BC = DA, it indicates that the opposite sides have the same lengths.
Step 3: Calculate slopes of opposite sides.
- Calculate the slope of side AB, BC, CD, and DA using the slope formula.
Step 4: Check if opposite sides have the same slopes.
- Compare the slopes of AB and CD, and BC and DA. If AB and CD have the same slope, and BC and DA have the same slope, it indicates that the opposite sides are parallel.
If both the opposite sides have the same lengths and the opposite sides have the same slopes, then the figure formed by the given vertices is a parallelogram.