Question

triangles jkl and jmn are shown on the grid below.
are △jkl and △jmn similar?
choose one option from each drop - down menu to answer the question and explain your reasoning.
triangles are similar if the ratio of the corresponding side lengths are choose... so, △jkl choose... similar to △jmn.
the ratio of jk to its corresponding side of △jmn is choose...
the ratio of kl to its corresponding side of △jmn is choose...

Explanation:

Step1: Calculate side lengths

Use the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\) (or count grid units for vertical/horizontal sides).
For \(MN\): vertical side, length \(= 2\) units.
For \(JK\): horizontal side, length \(= 2\) units.
For \(KL\): vertical side, length \(= 1\) unit.
For \(JM\): horizontal side, length \(= 4\) units.
For \(JN\): use distance formula (counting grid - \(x\) change \(= 4\), \(y\) change \(= 2\)), length \(=\sqrt{4^{2}+2^{2}}=\sqrt{20} = 2\sqrt{5}\).
For \(JL\): use distance formula (counting grid - \(x\) change \(= 2\), \(y\) change \(= 2\)), length \(=\sqrt{2^{2}+2^{2}}=\sqrt{8}=2\sqrt{2}\).

Step2: Check similarity ratio

Triangles are similar if \(\frac{JK}{JM}=\frac{KL}{MN}\).
\(\frac{JK}{JM}=\frac{2}{4}=\frac{1}{2}\), \(\frac{KL}{MN}=\frac{1}{2}\).

Answer:

Triangles are similar if the ratio of the corresponding side lengths are equal. So, \(\triangle{JKL}\) is similar to \(\triangle{JMN}\). The ratio of \(JK\) to its corresponding side of \(\triangle{JMN}\) is \(\frac{1}{2}\). The ratio of \(KL\) to its corresponding side of \(\triangle{JMN}\) is \(\frac{1}{2}\).