Question

1. are the triangles similar? yes or no? what is the similarity ratio? if there is no ratio, type none triangles are similar by complete the similarity statement. if not similar, type not similar. \\( \triangle t u v \sim \triangle \\)

Explanation:

Step1: Order the sides of each triangle

For $\triangle TUV$, the side lengths are 70, 91, 98. Let's order them: 70, 91, 98.
For $\triangle CAB$ (assuming the triangle is $\triangle CAB$ with sides 10, 13, 14), order the sides: 10, 13, 14.

Step2: Check the ratios of corresponding sides

Calculate the ratios of the longest sides: $\frac{98}{14} = 7$
Ratio of the middle sides: $\frac{91}{13} = 7$
Ratio of the shortest sides: $\frac{70}{10} = 7$

Since all three ratios are equal, the triangles are similar by the SSS (Side - Side - Side) similarity criterion.

Step3: Determine the similarity ratio

The similarity ratio is the ratio of the sides of $\triangle TUV$ to $\triangle CAB$ (or vice - versa, but we'll take the ratio of the larger to the smaller triangle here). Since each ratio is 7, the similarity ratio is 7 (or $\frac{7}{1}$).

Step4: Write the similarity statement

Since the order of the sides corresponds (shortest to shortest, middle to middle, longest to longest), $\triangle TUV \sim \triangle CAB$ (wait, no, let's check the side correspondences. Wait, in $\triangle TUV$: $TU = 70$, $UV = 98$, $TV = 91$. In $\triangle CAB$: $CA = 13$, $AB = 14$, $CB = 10$? Wait, no, I think I misassigned the triangle labels. Wait, let's re - assign. Let's list the sides of $\triangle TUV$: $TU = 70$, $UV = 98$, $TV = 91$. Sides of the other triangle: $CA = 13$, $AB = 14$, $CB = 10$. Wait, no, maybe the other triangle is $\triangle ACB$? Wait, no, let's do the ratio correctly. Let's take the sides of $\triangle TUV$: 70, 91, 98. Sides of the second triangle: 10, 13, 14. So 70/10 = 7, 91/13 = 7, 98/14 = 7. So the correspondence is $TU$ (70) corresponds to the side of length 10 (let's say $CB$), $TV$ (91) corresponds to $CA$ (13), and $UV$ (98) corresponds to $AB$ (14). So the similarity statement is $\triangle TUV \sim \triangle CBA$? Wait, no, let's check the labels again. Wait, the second triangle has vertices $C$, $B$, $A$ with $CB = 10$, $BA = 14$, $CA = 13$. The first triangle has $T$, $U$, $V$ with $TU = 70$, $UV = 98$, $TV = 91$. So $TU = 70$ (shortest side of $\triangle TUV$) corresponds to $CB = 10$ (shortest side of the second triangle), $TV = 91$ (middle side of $\triangle TUV$) corresponds to $CA = 13$ (middle side of the second triangle), and $UV = 98$ (longest side of $\triangle TUV$) corresponds to $BA = 14$ (longest side of the second triangle). So the similarity statement is $\triangle TUV \sim \triangle CBA$? Wait, no, the order of the letters should correspond to the order of the sides. So $T$ corresponds to $C$, $U$ corresponds to $B$, $V$ corresponds to $A$? Wait, no, $TU$ is between $T$ and $U$, $CB$ is between $C$ and $B$. So $T$ corresponds to $C$, $U$ corresponds to $B$, $V$ corresponds to $A$. Then $TU$ (70) corresponds to $CB$ (10), $UV$ (98) corresponds to $BA$ (14), $TV$ (91) corresponds to $CA$ (13). So $\triangle TUV \sim \triangle CBA$.

Answer:

• Are the triangles similar? yes
• What is the similarity ratio? 7
• Triangles are similar by SSS similarity
• Complete the similarity statement: $\triangle TUV \sim \triangle CBA$