Question

Triangle ABC has vertices at A(4, 2), B(1, 2), and C(4, 6) in the coordinate plane. The triangle will be rotated 270∘ counterclockwise around the point (-5, -7) and then rotated over = 0. What are the vertices of triangle A'B'C'?
A. A'(14, 2), B'(14, -1), C'(18, 2)


B. A'(0, 8), B'(0, 11), C'(-4, 8)


C. A'(-4, -16), B'(-4, -13), C'(-8, -16)


D. A'(4, 16), B'(4, 13), C'(8, 16)

Answer 1

Answer:

D.A'(4,16),B'(4,13),C'(8,16)

Step-by-step explanation:

We are given that triangle ABC has vertices at A(4,2),B(1,2) and C(4,6) in the coordinate plane.

First we change the given points

$A_1=(4+5,2+7)=(9,9)$

$B_1=(1+5,2+7)=(6,9)$

$C_1=(4+5,6+7)=(9,13)$

The rule of transformation of 270 degree rotation counterclockwise about origin is given by

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

After rotation 270 degrees about origin

Then, $A_2=(9,-9)$

$B_2=(9,-6)$

$C_2=(13,-9)$

Apply the rule

$(x,y)\rightarrow (x-5, y-7)$

$A_3=(4,-16)$

$B_3=(4,-13)$

$C_3=(8,-16)$

Hence, the coordinates of triangle ABC after 270 degrees counterclockwise rotation around (-5,-7) is given by

$A_3=(4,-16),B_3=(4,-13),C_3=(8,-16)$

After rotation y=0

The rule of transformation about y=0  is given by

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

Apply this rule then we get

$A'=(4,16)$

$B'=(4,13)$

$C'=(8,16)$

Hence, option D is true.