Answer:
[tex]A^{'}[/tex] = (1,[tex]\frac{5}{2}[/tex]), [tex]B^{'}[/tex] = ([tex]\frac{3}{2}[/tex],-1) and [tex]C^{'} = (2,2)[/tex]
OA = [tex]\sqrt[]{29}[/tex] units, OB = [tex]\sqrt[]{13}[/tex] units and OC = [tex]\sqrt[]{32}[/tex] units
O[tex]A^{'}[/tex] = [tex]\frac{\sqrt{29} }{2}[/tex] units,
O[tex]B^{'}[/tex] = [tex]\frac{\sqrt{13} }{2}[/tex] units and
O[tex]C^{'}[/tex] = [tex]\frac{\sqrt{32} }{2}[/tex] units
Step-by-step explanation:
Given vertices are A(2,5), B(3,-2), C(4,4) and the new vertices after dilation are [tex]A^{'}, B^{'}, C^{'}[/tex]
a.
Triangle needs to be dilated about the origin.
⇒The centre of dilation is the origin O(0,0).
scaling factor is [tex]\frac{1}{2}[/tex]
[tex]\frac{OA^{'}}{OA}[/tex] = [tex]\frac{OB^{'}}{OB}[/tex] = [tex]\frac{OC^{'}}{OC}[/tex] = [tex]\frac{1}{2}[/tex]
⇒ [tex]A^{'}, B^{'} and C^{'}[/tex] divide OA, OB and OC in the ratio 1:1 respectively.
Using the formula for the coordinates of a point dividing two points (0,0), (x,y) in the ratio 1:1 is ([tex]\frac{x}{2}[/tex],[tex]\frac{y}{2}[/tex]) :
[tex]A^{'}[/tex] = (1,[tex]\frac{5}{2}[/tex]), [tex]B^{'}[/tex] = ([tex]\frac{3}{2}[/tex],-1) and [tex]C^{'} = (2,2)[/tex]
b.
Using the formula for the distance between the points (0,0) and (x,y) is [tex]\sqrt[]{x^{2 }+ y^{2} }[/tex] :
OA = [tex]\sqrt[]{2^{2} +5^{2}}[/tex] = [tex]\sqrt[]{29}[/tex] units
OB = [tex]\sqrt[]{3^{2} +-2^{2}}[/tex] = [tex]\sqrt[]{13}[/tex] units
OC = [tex]\sqrt[]{4^{2} +4^{2}}[/tex] = [tex]\sqrt[]{32}[/tex] units
c.
From the property of dilation :
[tex]\frac{OA^{'}}{OA}[/tex] = [tex]\frac{OB^{'}}{OB}[/tex] = [tex]\frac{OC^{'}}{OC}[/tex] = [tex]\frac{1}{2}[/tex]
⇒O[tex]A^{'}[/tex] = [tex]\frac{OA}{2}[/tex] = [tex]\frac{\sqrt{29} }{2}[/tex] units
O[tex]B^{'}[/tex] = [tex]\frac{OB}{2}[/tex] = [tex]\frac{\sqrt{13} }{2}[/tex] units
O[tex]C^{'}[/tex] = [tex]\frac{OC}{2}[/tex] = [tex]\frac{\sqrt{32} }{2}[/tex] units
d.
Using the formula for the distance between two points :
O[tex]A^{'}[/tex] = [tex]\sqrt{1^{2} + (\frac{5}{2})^{2} }[/tex] = [tex]\frac{\sqrt{29} }{2}[/tex] units
O[tex]B^{'}[/tex] = [tex]\sqrt{(\frac{3}{2})^{2} + -1^{2}}[/tex] = [tex]\frac{\sqrt{13} }{2}[/tex] units
O[tex]C^{'}[/tex] = [tex]\sqrt[]{2^{2} +2^{2}}[/tex] = [tex]\frac{\sqrt{32} }{2}[/tex] units
which are the same as we predicted in part c.
