Question

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 1

The coordinates of E are
`E(x,y) = \left((9)/(2), 0 \right)`.

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:

`P'(x,y) = O(x,y) + k\cdot [P(x,y) - O(x,y)]` (1)

Where:

`O(x,y)` - Point of reference.

`P(x,y)` - Original point.

`P'(x,y)` - Dilated point.

`k` - Dilation factor.

By applying this operation, point B becomes point D:

`B(x,y) = (0,4)`,
`D(x,y) = (0,6)`

`D(x,y) = (0,0) + k\cdot [(0,4)- (0,0)]`

`D(x,y) = (0,0) + k\cdot (0,4)`

`(0,6) = (0,0) +(0,4\cdot k)`

`(0, 6) = (0,4\cdot k)`

`k = (3)/(2)`

Lastly, we transform point C into point E by applying the same operation:
`C(x,y) = (3, 0)`,
`O(x,y) = (0,0)` and
`k = (3)/(2)`

`E(x,y) = (0,0) + (3)/(2)\cdot [(3,0)-(0,0)]`

`E(x,y) = \left((9)/(2), 0 \right)`

The coordinates of E are
`E(x,y) = \left((9)/(2), 0 \right)`.