Answer:
Q1: The correct option is: 16
Q2: The correct options are: [tex]\frac{AB}{DE}=\frac{AC}{DF}[/tex] and [tex]\frac{AC}{DF}=\frac{BC}{EF}[/tex]
Q3: The correct option is: [tex]\overline{BC}=12; \overline{EF}=16[/tex]
Step-by-step explanation:
Question 1:
As here [tex]\triangle RST\sim \triangle MNO[/tex], so the ratio of the corresponding sides will be equal. That means.....
[tex]\frac{RS}{MN}=\frac{RT}{MO}\\ \\ \frac{8}{x}=\frac{6.5}{13}\\ \\ 6.5x=8*13\\ \\ x=\frac{8*13}{6.5}=16[/tex]
So, the length of the side [tex]x[/tex] will be 16.
Question 2:
If two triangles are similar, then the ratio of their corresponding sides should be equal. So, here the ratios of the corresponding sides are............
[tex]\frac{AB}{DE}=\frac{1}{3} \\ \\ \frac{BC}{EF}=\frac{2}{6}=\frac{1}{3}\\ \\ \frac{AC}{DF}=\frac{2}{7}[/tex]
So we can see that the ratio of side [tex]AC[/tex] and side [tex]DF[/tex] is not equal with the other ratios.
Thus, the proportions that show the triangles are not similar: [tex]\frac{AB}{DE}=\frac{AC}{DF}[/tex] and [tex]\frac{AC}{DF}=\frac{BC}{EF}[/tex]
Question 3:
Given that, [tex]\frac{AB}{DE}=\frac{BC}{EF}[/tex]
The ratio of [tex]AB[/tex] and [tex]DE[/tex] is given as [tex]\frac{3}{4}[/tex]
So, the ratio of [tex]BC[/tex] and [tex]EF[/tex] will be also [tex]\frac{3}{4}[/tex]
Among the four options, if [tex]BC=12[/tex] and [tex]EF=16[/tex], only then the ratio will be [tex]\frac{3}{4}[/tex]
[tex]\frac{BC}{EF} =\frac{12}{16}= \frac{3}{4}[/tex]
So, the lengths of [tex]BC[/tex] and [tex]EF[/tex] could be 12 and 16 respectively.
