Use algebra to classify quadrilateral JKLM. Select all the apply.
-Trapezoid
-Kite
-Square
-Rhombus
-Rectangle
-Parallelogram
Point J (2,1)
Point K (8,-1)
Point L (2,-6)
Point M (-4,-4)
To classify quadrilateral JKLM using algebra, we can start by finding the slopes of the sides and the lengths of the sides.
1. Finding the slopes of the sides:
The slope of a line passing through two points (x1, y1) and (x2, y2) is given by the formula:
m = (y2 - y1) / (x2 - x1)
Slope JK:
m1 = (-1 - 1) / (8 - 2) = -2/6 = -1/3
Slope KL:
m2 = (-6 - (-1)) / (2 - 8) = -5/(-6) = 5/6 = 0.833...
Slope LM:
m3 = (-4 - (-6)) / (-4 - 2) = 2/(-6) = -1/3
Slope MJ:
m4 = (1 - (-4)) / (2 - (-4)) = 5/6 = 0.833...
2. Finding the lengths of the sides:
Using the distance formula, the length of a line segment between two points (x1, y1) and (x2, y2) is given by the formula:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Length JK:
d1 = sqrt((8 - 2)^2 + (-1 - 1)^2) = sqrt(6^2 + (-2)^2) = sqrt(36 + 4) = sqrt(40) = 2sqrt(10)
Length KL:
d2 = sqrt((2 - 8)^2 + (-6 - (-1))^2) = sqrt((-6)^2 + (-5)^2) = sqrt(36 + 25) = sqrt(61)
Length LM:
d3 = sqrt((-4 - 2)^2 + (-4 - (-6))^2) = sqrt((-6)^2 + 2^2) = sqrt(36 + 4) = sqrt(40) = 2sqrt(10)
Length MJ:
d4 = sqrt((1 - (-4))^2 + (2 - (-6))^2) = sqrt(5^2 + 8^2) = sqrt(25 + 64) = sqrt(89)
Now we can analyze the characteristics of the quadrilateral JKLM based on the slopes and lengths:
- Trapezoid: The quadrilateral has one pair of parallel sides (JK and LM) due to the slopes being equal (-1/3 and -1/3). However, the lengths of the opposite sides (2sqrt(10) and 2sqrt(10)) are not equal, so it is not a trapezoid.
- Kite: A kite has two pairs of adjacent congruent sides. In this case, JK = LM = 2sqrt(10), and KL = MJ = sqrt(61) and sqrt(89) respectively. Therefore, JKLM is not a kite.
- Square: A square has four congruent sides and four right angles. Since the lengths of the sides are not equal, it is not a square.
- Rhombus: A rhombus has four congruent sides. Since the lengths of the sides are not equal, it is not a rhombus.
- Rectangle: A rectangle has four right angles. Since we don't have enough information about the angles, we cannot determine if JKLM is a rectangle.
- Parallelogram: A parallelogram has two pairs of parallel sides. In this case, JK and LM have equal slopes (-1/3 and -1/3), and KL and MJ have equal slopes (0.833... and 0.833...). So, JKLM is a parallelogram.
Therefore, the correct classifications for quadrilateral JKLM are:
- Parallelogram