Question

the graph shows triangles lmn and opq. is lmn similar to opq? justify your answer

Explanation:

Step1: Find the lengths of the sides of triangle \(LMN\)

Use the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
Assume \(L(-8,-4)\), \(M(-6,-8)\), \(N(-2,-2)\)
\(LM=\sqrt{(-6 + 8)^2+(-8 + 4)^2}=\sqrt{4 + 16}=\sqrt{20}=2\sqrt{5}\)
\(MN=\sqrt{(-2 + 6)^2+(-2 + 8)^2}=\sqrt{16 + 36}=\sqrt{52}=2\sqrt{13}\)
\(LN=\sqrt{(-2 + 8)^2+(-2 + 4)^2}=\sqrt{36+4}=\sqrt{40} = 2\sqrt{10}\)

Step2: Find the lengths of the sides of triangle \(OPQ\)

Assume \(O(3,-7)\), \(P(8,-6)\), \(Q(1,-3)\)
\(OP=\sqrt{(8 - 3)^2+(-6 + 7)^2}=\sqrt{25 + 1}=\sqrt{26}\)
\(PQ=\sqrt{(1 - 8)^2+(-3 + 6)^2}=\sqrt{49+9}=\sqrt{58}\)
\(OQ=\sqrt{(1 - 3)^2+(-3 + 7)^2}=\sqrt{4 + 16}=\sqrt{20}=2\sqrt{5}\)

Step3: Check the ratio of the sides

\(\frac{LM}{OQ}=\frac{2\sqrt{5}}{2\sqrt{5}} = 1\), \(\frac{MN}{OP}=\frac{2\sqrt{13}}{\sqrt{26}}=\sqrt{2}\), \(\frac{LN}{PQ}=\frac{2\sqrt{10}}{\sqrt{58}}=\sqrt{\frac{40}{58}}=\sqrt{\frac{20}{29}}\)
Since the ratios of the corresponding sides are not equal.

Answer:

\(\triangle LMN\) is not similar to \(\triangle OPQ\) because the ratios of their corresponding side lengths are not equal.