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Starrysoul100

[tex]\bold{\huge{\underline{ Solution \:1}}}[/tex]

  • Yes, ΔEDF and ΔBCA are similar

If we look at the both the triangles then the

  • [tex]\sf{ {\angle} E = {\angle}B }[/tex]
  • [tex]\sf{ {\angle} F = {\angle}A }[/tex]

From above, we can conclude that :-

Both the triangles are similar by AA Similarity

Hence, Option A is correct.

[tex]\bold{\huge{\underline{ Solution \:2}}}[/tex]

Let consider the given triangle be ABC and the line that divides the triangle consider it as DE

  • Here, DE || BC

So,

  • [tex]\sf{\dfrac{ AD }{ AB }}{\sf{ = }}{\sf{\dfrac{AE}{AC}}}{\sf{=}}{\sf{\dfrac{DE}{BC}}}[/tex]

Therefore,

  • ΔABC similar to ΔADE by SSS congruence similarity criterion.

Now,

We have to find the value of x

In ΔABC, By using BPT theorem

  • If the line is drawn parallel to one side of the triangle which intersect the other two sides at specific points then the other two sides are in proportion .

That is,

[tex]\sf{\dfrac{ AD }{ AB }}{\sf{ = }}{\sf{\dfrac{DE}{BC}}}[/tex]

Subsitute the required values,

[tex]\sf{\dfrac{ 6 }{ 6 + 4 }}{\sf{ = }}{\sf{\dfrac{8}{x}}}[/tex]

[tex]\sf{\dfrac{ 6 }{ 10 }}{\sf{ = }}{\sf{\dfrac{8}{x}}}[/tex]

[tex]\sf{\dfrac{ 3 }{ 5 }}{\sf{ = }}{\sf{\dfrac{8}{x}}}[/tex]

[tex]{\sf{ x = 8{\times} }}{\sf{\dfrac{5}{3}}}[/tex]

[tex]{\sf{ x = }}{\sf{\dfrac{40}{3}}}[/tex]

[tex]{\sf{ x = 13}}{\sf{\dfrac{1}{3}}}[/tex]

Hence, Option C is correct

[tex]\bold{\huge{\underline{ Solution \:3}}}[/tex]

Here,

  • SRT similar to BAC

For x,

By using BPT theorem,

[tex]\sf{\dfrac{ AB }{ RS }}{\sf{ = }}{\sf{\dfrac{BC}{ST}}}[/tex]

Subsitute the required values,

[tex]\sf{\dfrac{ 2 }{ 4 }}{\sf{ = }}{\sf{\dfrac{x}{6}}}[/tex]

[tex]\sf{\dfrac{ 1 }{ 2 }}{\sf{ = }}{\sf{\dfrac{x}{6}}}[/tex]

[tex]\sf{ x = }{\sf{\dfrac{6}{2}}}[/tex]

[tex]\sf{ x = 3 }[/tex]

Thus, The value of x is 3

For y

Again by using BPT theorem,

[tex]\sf{\dfrac{ AB }{ RS }}{\sf{ = }}{\sf{\dfrac{AC}{RT}}}[/tex]

Subsitute the required values,

[tex]\sf{\dfrac{ 2 }{ 4 }}{\sf{ = }}{\sf{\dfrac{5}{y}}}[/tex]

[tex]\sf{\dfrac{ 1 }{ 2 }}{\sf{ = }}{\sf{\dfrac{5}{y}}}[/tex]

[tex]\sf{ y = 5{\times}2 }[/tex]

[tex]\sf{ y = 10 }[/tex]

Thus, The value of y is 10

Hence, Option A is correct.

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