A stained glass window is graphed on a coordinate plane with vertices at W (8,8) , X (9,4) , Y (2,−2) , and Z (4,7) A line through the longer diagonal has the equation y=(5)/(3)x−(16)/(3) How can midpoints be used to help determine whether the shape is a kite?
Answer
A
The midpoint of line segment cap x cap z is (6.5, 5.5), which is a solution to the equation for the longer diagonal, confirming a property of a kite.
B
The midpoint of line segment cap w cap y is (5, 3), which is a solution to the equation for the longer diagonal, so the shape cannot be a kite.
C
The midpoint of line segment cap x cap z is (6.5, 5.5), which is a solution to the equation for the longer diagonal, so the shape cannot be a kite.
D
The midpoint of line segment cap w cap y is (5, 3), which is a solution to the equation for the longer diagonal, confirming a property of a kite.
Explanation
The longer diagonal is WY, since its equation is the given line. The midpoint of XZ is ((9 + 4)/2, (4 + 7)/2) = (6.5, 5.5). Substituting x = 6.5 into the line gives y = (5/3)(6.5) − 16/3 = 5.5, so the midpoint of XZ lies on WY. This shows WY bisects XZ, a property of a kite, so choice A is correct.
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