Which of the following are true attributes about the quadrilateral that has vertices at K(2,5), L(9,−2), M(1,−5), and N(−6,2)?
I. The opposite sides are parallel
II. The opposite sides are congruent
III. The diagonals are congruent
IV. The diagonals are perpendicular
Iv and I
To determine the attributes of the quadrilateral with vertices at K(2,5), L(9,−2), M(1,−5), and N(−6,2), we can analyze the properties of quadrilaterals using the coordinate geometry.
I. The opposite sides are parallel:
To check if the opposite sides are parallel, we can analyze the slopes of the sides.
Slope of KL = (y2 - y1) / (x2 - x1) = (-2 - 5) / (9 - 2) = -7/7 = -1
Slope of MN = (2 - (-5)) / (-6 - 1) = 7 / -7 = -1
As both slopes are equal (-1), the opposite sides KL and MN are parallel.
II. The opposite sides are congruent:
To check if the opposite sides are congruent, we can analyze the lengths of the sides.
Distance between KL = sqrt((9 - 2)^2 + (-2 - 5)^2) = sqrt(49 + 49) = sqrt(98)
Distance between MN = sqrt((-6 - 1)^2 + (2 - (-5))^2) = sqrt(49 + 49) = sqrt(98)
As both distances are equal (sqrt(98)), the opposite sides KL and MN are congruent.
III. The diagonals are congruent:
To check if the diagonals are congruent, we can analyze the lengths of the diagonals.
Distance between KM = sqrt((1 - 2)^2 + (-5 - 5)^2) = sqrt(1 + 100) = sqrt(101)
Distance between LN = sqrt((-6 - 9)^2 + (2 - (-2))^2) = sqrt(225 + 16) = sqrt(241)
As the distances between KM and LN are different (sqrt(101) and sqrt(241) respectively), the diagonals are not congruent.
IV. The diagonals are perpendicular:
To check if the diagonals are perpendicular, we can analyze the slopes of the diagonals.
Slope of KM = (-5 - 5) / (1 - 2) = -10 / -1 = 10
Slope of LN = (2 - (-2)) / (-6 - 9) = 4 / -15 = -4/15
As the slopes of KM and LN are not negative reciprocals of each other (10 and -4/15), the diagonals are not perpendicular.
Based on the above calculations:
I. The statement "The opposite sides are parallel" is true.
II. The statement "The opposite sides are congruent" is true.
III. The statement "The diagonals are congruent" is false.
IV. The statement "The diagonals are perpendicular" is false.
To determine which of the given attributes are true about the quadrilateral with the given vertices, we need to analyze its properties.
I. To determine whether the opposite sides are parallel, we can compare the slopes of each pair of opposite sides. If the slopes are equal, then the sides are parallel.
II. To determine whether the opposite sides are congruent, we can calculate the distances between each pair of opposite vertices. If the distances are equal, then the sides are congruent.
III. To determine whether the diagonals are congruent, we can calculate the lengths of both diagonals. If the lengths are equal, then the diagonals are congruent.
IV. To determine whether the diagonals are perpendicular, we can calculate the slopes of the lines passing through each diagonal. If the slopes are negative reciprocals of each other, then the diagonals are perpendicular.
Let's now perform the required calculations to find the answer:
1. Calculate the slopes of KL, LM, MN, and NK:
Slope of KL = (−2−5)/(9−2) = −7/7 = −1
Slope of LM = (−5−(−2))/(1−9) = −3/−8 = 3/8
Slope of MN = (2−(−5))/((−6)−1) = 7/−7 = −1
Slope of NK = (5−2)/(2−(−6)) = 3/8
As the slopes of KL and MN are equal, and the slopes of LM and NK are equal, we can conclude that the opposite sides KL and MN, as well as LM and NK, are parallel. Hence, I is true.
2. Calculate the distances between KL, LM, MN, and NK:
Distance between KL = sqrt((9−2)^2 + (−2−5)^2) = sqrt(49 + 49) = sqrt(98)
Distance between LM = sqrt((1−9)^2 + (−5−(−2))^2) = sqrt(64 + 9) = sqrt(73)
Distance between MN = sqrt((−6−1)^2 + (2−(−5))^2) = sqrt(49 + 49) = sqrt(98)
Distance between NK = sqrt((2−(−6))^2 + (5−2)^2) = sqrt(64 + 9) = sqrt(73)
Since the distances between KL and MN, as well as LM and NK, are equal, we can conclude that the opposite sides KL and MN, as well as LM and NK, are congruent. Hence, II is true.
3. Calculate the lengths of the diagonals KM and LN:
Length of KM = sqrt((1−2)^2 + (−5−5)^2) = sqrt(1 + 100) = sqrt(101)
Length of LN = sqrt((−6−9)^2 + (2−(−2))^2) = sqrt(225 + 16) = sqrt(241)
As the lengths of KM and LN are not equal, we can conclude that the diagonals KM and LN are not congruent. Hence, III is false.
4. Calculate the slopes of the lines passing through KM and LN:
Slope of line KM = (−5−2)/(1−2) = −7/−1 = 7
Slope of line LN = (2−(−2))/((−6)−9) = 4/−15 = −4/15
As the slopes of KM and LN are not the negative reciprocals of each other, we can conclude that the diagonals KM and LN are not perpendicular. Hence, IV is false.
To summarize the answers:
I. True - The opposite sides are parallel.
II. True - The opposite sides are congruent.
III. False - The diagonals are not congruent.
IV. False - The diagonals are not perpendicular.
check the side lengths and the slopes
what do you find?