Question

the vertices of δrst 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 δrst?
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 \(\triangle RST\)

Use the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
For \(RS\): \(x_1=-1,y_1 = - 1,x_2=-1,y_2 = 11\), then \(RS=\sqrt{(-1+1)^2+(11 + 1)^2}=\sqrt{0+144}=12\).
For \(ST\): \(x_1=-1,y_1 = 11,x_2=4,y_2 = 11\), then \(ST=\sqrt{(4 + 1)^2+(11-11)^2}=\sqrt{25+0}=5\).
For \(RT\): \(x_1=-1,y_1 = - 1,x_2=4,y_2 = 11\), then \(RT=\sqrt{(4 + 1)^2+(11 + 1)^2}=\sqrt{25 + 144}=\sqrt{169}=13\).
The side - lengths of \(\triangle RST\) are \(5\), \(12\), and \(13\) units.

Step2: Check the similarity ratio

If a triangle is similar to \(\triangle RST\), its side - lengths should be in proportion to \(5\), \(12\), \(13\).
For a set of side - lengths \(10\), \(24\), \(26\), we can check the ratios: \(\frac{10}{5}=2\), \(\frac{24}{12}=2\), \(\frac{26}{13}=2\).

Answer:

\(10\), \(24\), and \(26\) units