Question

the vertices of \\( \triangle rst \\) are \\( r ( - 1, - 1 ) \\), \\( s ( - 1,11 ) \\) and \\( t ( 4,11 ) \\). which could be the side lengths of a triangle that is similar but not congruent to \\( \triangle rst \\)?
10, 12, and 13 units
5, 24, and 26 units
10, 24, and 26 units
5, 12, and 13 units

Explanation:

Step1: Calculate the side lengths of \(\triangle{RST}\)

• For \(RS\): Using the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), with \(R(-1,-1)\) and \(S(-1,11)\). Here \(x_1=-1,y_1 = - 1,x_2=-1,y_2 = 11\). Then \(RS=\sqrt{(-1+1)^2+(11 + 1)^2}=\sqrt{0+(12)^2}=12\)
• For \(ST\): Using the distance formula with \(S(-1,11)\) and \(T(4,11)\). Here \(x_1=-1,y_1 = 11,x_2=4,y_2 = 11\). Then \(ST=\sqrt{(4 + 1)^2+(11-11)^2}=\sqrt{(5)^2+0}=5\)
• For \(RT\): Using the distance formula with \(R(-1,-1)\) and \(T(4,11)\). Then \(RT=\sqrt{(4 + 1)^2+(11 + 1)^2}=\sqrt{25+144}=\sqrt{169}=13\)

Step2: Check the similarity ratio

• A triangle similar but not congruent to \(\triangle{RST}\) (with side - lengths \(5,12,13\)) will have side - lengths that are a non - one multiple of \(5,12,13\)
• If we multiply each side - length of \(\triangle{RST}\) by \(2\), we get \(5\times2 = 10\), \(12\times2=24\), \(13\times2 = 26\)

Answer:

\(10,24,\text{and }26\text{ units}\)