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On a coordinate plane, triangle X Y Z is shown. Point X is at (1, 3), point Y is at (4, negative 1), and point Z is at (5, 6).
Which statement proves that △XYZ is an isosceles right triangle?
XZ is not perpendicular to XY
XZXY
The slope of XZ is Three-fourths, the slope of XY is Negative four-thirds, and XZ = XY = 5.
The slope of XZ is Three-fourths, the slope of XY is Negative four-thirds, and the slope of ZY = 7.
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"The slope of XZ is Three-fourths, the slope of XY is Negative four-thirds, and XZ = XY = 5."
Reason: XZ vector = (4,3) so slope = 3/4 and length = sqrt(4^2+3^2)=5. XY vector = (3,-4) so slope = -4/3 and length = sqrt(3^2+(-4)^2)=5. The slopes are negative reciprocals, so XZ ⟂ XY, and the two sides have equal length, so triangle XYZ is isosceles and has a right angle at X — an isosceles right triangle.
Reason: XZ vector = (4,3) so slope = 3/4 and length = sqrt(4^2+3^2)=5. XY vector = (3,-4) so slope = -4/3 and length = sqrt(3^2+(-4)^2)=5. The slopes are negative reciprocals, so XZ ⟂ XY, and the two sides have equal length, so triangle XYZ is isosceles and has a right angle at X — an isosceles right triangle.
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