Con Triangle ABC with vertices A(0,0), B(0,4), and C(3,0) is dilated to form triangle ADE. What is the E if D has coordinates (0,6)? 106

Answer:The coordinates of E are [tex]E(x,y) = \left(\frac{9}{2}, 0 \right)[/tex].Step-by-step explanation:The triangle ABC represents a right triangle as both sides AB and AC are orthogonal to each other. The side AB is in the y axis, whereas the side AC is in the x axis. The triangle is dilated with respect to the origin, in which point A is set. Vectorially speaking, dilation is defined by the following operation:[tex]P'(x,y) = O(x,y) + k\cdot [P(x,y) - O(x,y)][/tex] (1)Where:[tex]O(x,y)[/tex] - Point of reference.[tex]P(x,y)[/tex] - Original point.[tex]P'(x,y)[/tex] - Dilated point.[tex]k[/tex] - Dilation factor.By applying this operation, point B becomes point D: [tex]B(x,y) = (0,4)[/tex], [tex]D(x,y) = (0,6)[/tex][tex]D(x,y) = (0,0) + k\cdot [(0,4)- (0,0)][/tex][tex]D(x,y) = (0,0) + k\cdot (0,4)[/tex][tex](0,6) = (0,0) +(0,4\cdot k)[/tex][tex](0, 6) = (0,4\cdot k)[/tex][tex]k = \frac{3}{2}[/tex]Lastly, we transform point C into point E by applying the same operation: [tex]C(x,y) = (3, 0)[/tex], [tex]O(x,y) = (0,0)[/tex] and [tex]k = \frac{3}{2}[/tex][tex]E(x,y) = (0,0) + \frac{3}{2}\cdot [(3,0)-(0,0)][/tex][tex]E(x,y) = \left(\frac{9}{2}, 0 \right)[/tex]The coordinates of E are [tex]E(x,y) = \left(\frac{9}{2}, 0 \right)[/tex].