Mathematics: Crystal is amazed! she graphed δabc using the points a(5, –1), b(3, –7), and c(6, –2). then she rotated δabc 90° counterclockwise (↺) about the origin to find δa′b′c′. meanwhile, her teammate took a different triangle (δtuv) and rotated it 90° clockwise (↻) about the origin to find δt′u′v′. amazingly, δa′b′c′ and δt′u′v′ ended up having exactly the same points! name the coordinates of the vertices of δtuv.

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Crystal is amazed! she graphed δabc using the points

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andrealch13p3w89f

16.01.2022

Mathematics

Crystal graphed ΔABC using the following points:

$A(5, -1) \\ \\ B(3, -7) \\ \\ C(6, -2)$

On the coordinate plane, consider the point (x,y) . To rotate this point by 90° around the origin in counterclockwise direction, you can always swap the x- and y-coordinates and then multiply the new x-coordinate by -1. In a mathematical language this is as follows:

$(x,y)\rightarrow(-y,x)$

Therefore, for ΔA′B′C′ we have:

$\boxed{A'(1, 5)} \\ \\ \boxed{B'(7, 3)} \\ \\ \boxed{C'(2, 6)}$

Since ΔA′B′C′ and ΔT′U′V′ ended up having exactly the same points, then:

$T'(1, 5) \\ \\U'(7, 3) \\ \\V'(2, 6)$

On the other hand, in clockwise direction we have the following rule:

$(x,y)\rightarrow(y,-x)$

Therefore. we must find (x,y)

to get ΔTUV here, so:

$T'(y,-x)=T'(1, 5) \therefore x=-5 \ and \ y=1 \rightarrow T(-5,1)$

$T'(y,-x)=T'(1, 5) \therefore x=-5 \ and \ y=1 \rightarrow \boxed{T(-5,1)} \\ \\U'(y,-x)=U'(7, 3) \therefore x=-3 \ and \ y=7 \rightarrow \boxed{U(-3,7)} \\ \\V'(y,-x)=V'(2, 6) \therefore x=-6 \ and \ y=2 \rightarrow \boxed{V(-6,2)}$

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