Question

determine if triangle lmn and triangle opq are or are not similar, and, if they are, state how you know. (note that figures are not necessarily drawn to scale.)

triangle lmn has sides: n to m is 19, m to l is 34, l to n is 26.
triangle opq has sides: p to o is 104, o to q is 78, q to p is 57.

answer attempt 1 out of 3

the triangles dropdown similar.

Explanation:

Step1: Order the sides of each triangle

For $\triangle LMN$: Sides are $19$, $26$, $34$. Order them: $19$, $26$, $34$.
For $\triangle OPQ$: Sides are $57$, $78$, $104$. Order them: $57$, $78$, $104$.

Step2: Check the ratios of corresponding sides

Calculate the ratios:
$\frac{19}{57} = \frac{1}{3}$, $\frac{26}{78} = \frac{1}{3}$, $\frac{34}{104} = \frac{17}{52}$? Wait, no, recalculate $\frac{34}{104}$: Simplify, divide numerator and denominator by 2: $\frac{17}{52}$? Wait, no, $19\times3 = 57$, $26\times3 = 78$, $34\times3 = 102$? Wait, no, $34\times3 = 102$, but the third side of $\triangle OPQ$ is $104$. Wait, I made a mistake. Wait, re - order $\triangle LMN$ sides: $19$, $26$, $34$; $\triangle OPQ$: $57$, $78$, $104$. Let's check $\frac{19}{57}=\frac{1}{3}$, $\frac{26}{78}=\frac{1}{3}$, $\frac{34}{104}=\frac{17}{52}\approx0.3269$, and $\frac{1}{3}\approx0.3333$. Wait, maybe I ordered the sides wrong. Let's check again. Wait, maybe the sides of $\triangle LMN$: $19$, $26$, $34$; $\triangle OPQ$: $57$, $78$, $104$. Let's check $\frac{19}{57}=\frac{1}{3}$, $\frac{26}{78}=\frac{1}{3}$, $\frac{34}{104}=\frac{17}{52}$, but $19\times 3=57$, $26\times3 = 78$, $34\times3=102
eq104$. Wait, that's a mistake. Wait, maybe I mixed up the sides. Wait, maybe the sides of $\triangle LMN$ are $19$, $26$, $34$ and $\triangle OPQ$ are $57$, $78$, $104$. Let's check $\frac{19}{57}=\frac{1}{3}$, $\frac{26}{78}=\frac{1}{3}$, $\frac{34}{104}=\frac{17}{52}$, but $34\times3 = 102
eq104$. Wait, maybe I have the sides of the triangles wrong. Wait, no, maybe I made a calculation error. Wait, $34\div104=\frac{17}{52}\approx0.3269$, and $\frac{1}{3}\approx0.3333$. The difference is due to a miscalculation? Wait, no, $19\times3 = 57$, $26\times3=78$, $34\times3 = 102$. But the third side of $\triangle OPQ$ is $104$. So the ratios are not equal. Wait, but maybe I ordered the sides incorrectly. Let's try another order. Let's take the sides of $\triangle LMN$ as $19$, $34$, $26$ and $\triangle OPQ$ as $57$, $104$, $78$. Then $\frac{19}{57}=\frac{1}{3}$, $\frac{34}{104}=\frac{17}{52}$, $\frac{26}{78}=\frac{1}{3}$. Still, $\frac{34}{104}
eq\frac{1}{3}$. Wait, maybe the problem has a typo, or I made a mistake. Wait, no, let's check again. Wait, $19\times 3 = 57$, $26\times3 = 78$, $34\times3=102$. The third side of the second triangle is $104$, which is not $102$. So the ratios of the corresponding sides are not equal. So the triangles are not similar? Wait, but maybe I messed up the correspondence. Wait, maybe the sides of $\triangle LMN$: $19$, $26$, $34$; $\triangle OPQ$: $57$, $78$, $104$. Let's check $\frac{19}{57}=\frac{1}{3}$, $\frac{26}{78}=\frac{1}{3}$, $\frac{34}{104}=\frac{17}{52}$. Since $\frac{17}{52}
eq\frac{1}{3}$, the triangles are not similar. Wait, but maybe I made a mistake in the side lengths. Wait, the first triangle: $NM = 19$, $NL = 26$, $LM = 34$. The second triangle: $PO = 104$, $PQ = 57$, $QO = 78$. So let's list the sides in order from smallest to largest for each triangle. $\triangle LMN$: $19$ (smallest), $26$, $34$ (largest). $\triangle OPQ$: $57$ (smallest), $78$, $104$ (largest). Now, check the ratios of smallest to smallest, middle to middle, largest to largest. $\frac{19}{57}=\frac{1}{3}$, $\frac{26}{78}=\frac{1}{3}$, $\frac{34}{104}=\frac{17}{52}\approx0.3269$, and $\frac{1}{3}\approx0.3333$. Since $\frac{34}{104}
eq\frac{1}{3}$, the three ratios are not equal. So the triangles are not similar.

Answer:

The triangles are not similar. The reason is that when we check the ratios of the corresponding sides (after ordering the sides of each triangle from smallest to largest), the ratios $\frac{19}{57}=\frac{1}{3}$, $\frac{26}{78}=\frac{1}{3}$, and $\frac{34}{104}=\frac{17}{52}$ are not all equal, so the Side - Side - Side (SSS) similarity criterion is not satisfied.