Resolved: "Draw a graph for the original figure and..."

abondantQ

Answered question

2020-10-27

Draw a graph for the original figure and its dilated image. Check whether the dilation is a similarity transformation or not.
Given:
The given vertices are
original

\to A (2, 3), B (0, 1), C (3, 0)

image

\to D (4, 6), F (0, 2), G (6, 0)

Answer & Explanation

davonliefI

Skilled2020-10-28Added 79 answers

Calculation:
The graph for the points is given below.

Find the distance (using distance formula) of corresponding sides and find the ratio.
$A B = \sqrt{{(2 - 0)}^{2} + {(3 - 1)}^{2}} = \sqrt{8} = 2 \sqrt{2}$
$D F = \sqrt{{(4 - 0)}^{2} + {(6 - 2)}^{2}} = \sqrt{32} = 4 \sqrt{2}$
$\frac{A B}{D F} = \frac{2 \sqrt{2}}{4 \sqrt{2}} = \frac{1}{2}$
$A C = \sqrt{{(2 - 3)}^{2} + {(3 - 0)}^{2}} = \sqrt{10}$
$D G = \sqrt{{(4 - 6)}^{2} + {(6 - 0)}^{2}} = \sqrt{40} = 2 \sqrt{10}$
$\frac{A C}{D G} = \frac{\sqrt{10}}{2 \sqrt{10}} = \frac{1}{2}$
$B C = \sqrt{{(0 - 3)}^{2} + {(1 - 0)}^{2}} = \sqrt{10}$
$F G = \sqrt{{(0 - 6)}^{2} + {(2 - 0)}^{2}} = \sqrt{40} = 2 \sqrt{10}$
$\frac{B C}{F G} = \frac{\sqrt{10}}{2 \sqrt{10}} = \frac{1}{2}$
From the above result, the lenght of the sides are proportional.
By the use of SSS similarity,
$△ D F G \sim △ A B C$ .
Hence the dilation is a similarity transformation.

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