Triangle XYZ is shown on the coordinate plane.
A triangle on the coordinate plane with vertices X at 5 comma 6, Y at 10 comma 1, and Z at 2 comma 1.
If triangle XYZ is translated using the rule (x, y) → (x + 5, y − 3) and then rotated 90° clockwise to create triangle X″Y″Z″, what is the location of X″?
(−3, 10)
(−2, −15)
(−2, −7)
(3, −10)
After translating triangle XYZ using the rule (x, y) → (x + 5, y − 3), we get the new vertices: X' at (10, 3), Y' at (15, -2), and Z' at (7, -2).
To rotate the triangle 90° clockwise around the origin, we use the rotation matrix for a 90° clockwise rotation:
[ 0 1 ]
[-1 0 ]
Applying this rotation matrix to each of the new vertices (X', Y', Z') gives us:
X'' = (0 * 10 + 1 * 3, -1 * 10 + 0 * 3) = (3, -10)
Therefore, the location of X'' is (3, -10).
So the correct answer is: (3, -10)