## The vertices of ▲ A(2,-3), B(5,-5) and C(3,0). ▲ ABC is translated through vector (2,-3)' to get the image ▲A'B'C' and ▲A'B'C is rotated through quarter turn in anticlockwise direction about origin to get image ▲A"B''C''. Find the coordinates of the vertices of ▲A'B'C and ▲A''B''C'' and plot it in graph?

Mar 23, 2018

The vertices of ∆A''B''C'' are
$A ' ' \left(9 , 4\right) , B ' ' \left(8 , 7\right) \mathmr{and} C ' ' \left(3 , 5\right)$.

#### Explanation:

The vertices of ∆ABC are $A \left(2 , - 3\right) , B \left(5 , - 5\right) \mathmr{and} C \left(3 , 0\right)$.

Translation vector $= \left(2 , - 3\right)$

When the translation vector, $T = \left(a , b\right)$ then,

P(x, y) ⟶ P'(x+a, y+b)

 A(2, -3) ⟶ A'{2 + 2, -3 + (-3)} = A'(4, -9)
 B(5, -5) ⟶ B'{5 + 2, -5 + (-3)} = B'(7, -8)
 C(3, 0) ⟶ C'{3 +2, 0 + (-3)} = C'(5, -3)

The vertices of ∆A'B'C' are $A ' \left(4 , - 9\right) , B ' \left(7 , - 8\right) \mathmr{and} C ' \left(5 , - 3\right)$

Under rotation about origin through quarter turn in anticlockwise direction $\left(+ {90}^{0}\right)$,

P(x, y) → P'(-y,x)

 A'(4, -9) → A''(9, 4)
B'(7, -8) → B''(8, 7)
C'(5, -3) → C''(3, 5)
The vertices of ∆A''B''C'' are
$A ' ' \left(9 , 4\right) , B ' ' \left(8 , 7\right) \mathmr{and} C ' ' \left(3 , 5\right)$.

Here, yellow colored triangle represents ∆ABC with vertices $A \left(2 , - 3\right) , B \left(5 , - 5\right) \mathmr{and} C \left(3 , 0\right)$.

Red colored triangle represents ∆A'B'C' with vertices $A ' \left(4 , - 9\right) , B ' \left(7 , - 8\right) \mathmr{and} C ' \left(5 , - 3\right)$.

Blue colored triangle represents ∆A''B''C'' with vertices $A ' ' \left(9 , 4\right) , B ' ' \left(8 , 7\right) \mathmr{and} C ' ' \left(3 , 5\right)$.

Hope it helps :)