Answer:
The set of vertices that represents a kite based on the given distances is B) K(3,2), T(7,5), E(4,-1).
Option (B) is true.
Explanation:
To determine which set of vertices represents a kite, we need to check if the given distances satisfy the properties of a kite.
The properties of a kite are:
Two pairs of sides that have equal length.
One pair of diagonals that are perpendicular to each other.
Let's calculate the distances between the vertices of each set:
A)
K(1,3), T(5,7), E(2,1)
KT = √((5-1)² + (7-3)²) = √16 + 16 = √32
KE = √((2-1)² + (1-3)²) = √1 + 4 = √5
TE = √((5-2)² + (7-1)²) = √9 + 36 = √45
B)
K(3,2), T(7,5), E(4,-1)
KT = √((7-3)² + (5-2)²) = √16 + 9 = √25 = 5
KE = √((4-3)² + (-1-2)²) = √1 + 9 = √10
TE = √((7-4)² + (5+1)²) = √9 + 36 = √45
C)
K(0,4), T(6,2), E(3,-3)
KT = √((6-0)² + (2-4)²) = √36 + 4 = √40
KE = √((3-0)² + (-3-4)²) = √9 + 49 = √58
TE = √((6-3)² + (2+3)²) = √9 + 25 = √34
D)
K(2,-1), T(8,4), E(5,0)
KT = √((8-2)² + (4+1)²) = √36 + 25 = √61
KE = √((5-2)² + (0+1)²) = √9 + 1 = √10
TE = √((8-5)² + (4-0)²) = √9 + 16 = √25 = 5
Based on the distances calculated, only option B) K(3,2), T(7,5), E(4,-1) satisfies the properties of a kite.
The distances KT and TE are equal (both 5) and the distance KE is different, indicating two pairs of sides with equal lengths.
Therefore,
The correct answer is B) K(3,2), T(7,5), E(4,-1).