Question

question 1 (multiple choice worth 1 points) (03 02pr mc) the vertices of \\( \triangle ghi \\) are \\( g(2,4) \\), \\( h(4,8) \\), and \\( i(8,4) \\). the vertices of \\( \triangle jkl \\) are \\( j(1,1) \\), \\( k(2,3) \\), and \\( l(4,1) \\). which conclusion is true about the triangles? they are congruent by the definition of congruence in terms of rigid motions they are similar by the definition of similarity in terms of a dilation the ratio of their corresponding sides is \\( 1:3 \\) the ratio of their corresponding angles is \\( 1:3 \\)

Explanation:

Step1: Calculate the lengths of the sides of \(\triangle GHI\)

Use the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
For \(GH\): \(x_1 = 2,y_1 = 4,x_2 = 4,y_2 = 8\)
\(GH=\sqrt{(4 - 2)^2+(8 - 4)^2}=\sqrt{4 + 16}=\sqrt{20}=2\sqrt{5}\)
For \(HI\): \(x_1 = 4,y_1 = 8,x_2 = 8,y_2 = 4\)
\(HI=\sqrt{(8 - 4)^2+(4 - 8)^2}=\sqrt{16 + 16}=\sqrt{32}=4\sqrt{2}\)
For \(GI\): \(x_1 = 2,y_1 = 4,x_2 = 8,y_2 = 4\)
\(GI=\sqrt{(8 - 2)^2+(4 - 4)^2}=\sqrt{36}=6\)

Step2: Calculate the lengths of the sides of \(\triangle JKL\)

For \(JK\): \(x_1 = 1,y_1 = 1,x_2 = 2,y_2 = 3\)
\(JK=\sqrt{(2 - 1)^2+(3 - 1)^2}=\sqrt{1 + 4}=\sqrt{5}\)
For \(KL\): \(x_1 = 2,y_1 = 3,x_2 = 4,y_2 = 1\)
\(KL=\sqrt{(4 - 2)^2+(1 - 3)^2}=\sqrt{4 + 4}=\sqrt{8}=2\sqrt{2}\)
For \(JL\): \(x_1 = 1,y_1 = 1,x_2 = 4,y_2 = 1\)
\(JL=\sqrt{(4 - 1)^2+(1 - 1)^2}=\sqrt{9}=3\)

Step3: Check the ratio of corresponding sides

\(\frac{GH}{JK}=\frac{2\sqrt{5}}{\sqrt{5}} = 2\), \(\frac{HI}{KL}=\frac{4\sqrt{2}}{2\sqrt{2}}=2\), \(\frac{GI}{JL}=\frac{6}{3}=2\).
Since the ratio of corresponding sides is \(2\), and angles of similar triangles are equal (ratio of angles is \(1:1\)).
If we consider a dilation (a similarity transformation) with a scale factor of \(2\) (or \(\frac{1}{2}\) depending on the direction of mapping), \(\triangle GHI\) and \(\triangle JKL\) are similar.

Answer:

They are similar by the definition of similarity in terms of a dilation.